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4th Std Scholarship · Intelligence Test

Series

5 concepts · 40 practice questions with answers · Free to read

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Series with the same difference

From daily life

Aarav puts Rs 5 in his money box every day. The amount grows 5, 10, 15, 20. Each time the same 5 is added. So tomorrow the box will have Rs 25.

Simple definition

A row of numbers is a series and each number in it is a term. The gap between two neighbouring terms is the difference. Difference = next term − earlier term. If all the differences are the same, the series is regular.

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Series with the same difference

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How to find the difference

SeriesDifferenceNext term
7, 12, 17, 22+ 527
50, 44, 38, 32− 626
4, 15, 26, 37+ 1148
1
Rule 1

Find the difference between neighbouring terms (next term − earlier term) and check that all the differences are the same.

  • e.g. 7, 12, 17, 22 : 12 − 7 = 5, 17 − 12 = 5, 22 − 17 = 5. The difference is the same.
2
Rule 2

If the difference is the same, add it to the last term. If the numbers are getting smaller, the difference is negative (−), so the term goes down.

  • e.g. 50, 44, 38, 32 : the difference is −6, so the next term is 32 − 6 = 26.
3
Rule 3

Which term = first term + (position − 1) × difference. This gives a far-away term without writing the terms in between.

  • e.g. the 10th term of 7, 12, 17, … = 7 + 9 × 5 = 52.
4
Rule 4

If a term in the middle is missing, add the two terms on its sides and divide by 2.

  • e.g. 6, 13, ?, 27, 34 : (13 + 27) ÷ 2 = 20.
Example 1 · easy

What is the next term of 4, 9, 14, 19, ? ?

  1. Difference = 9 − 4 = 5.
  2. Next term = 19 + 5 = 24.
Example 2 · medium

What is the next term of 90, 83, 76, 69, ? ?

  1. Difference = 83 − 90 = −7 (the numbers are going down).
  2. Next term = 69 − 7 = 62.
Example 3 · exam type

What is the 25th term of the series 3, 8, 13, 18, … ?

  1. Difference = 8 − 3 = 5.
  2. 25th term = 3 + 24 × 5 = 3 + 120 = 123.
  3. Check : the same rule gives the 5th term = 3 + 4 × 5 = 23, and 18 + 5 = 23 as well. They match.
Remember

Always find the difference as "next − earlier". If the answer is negative (−), the series is going down.

Common mistake

Fixing the rule from the first two terms only. In 2, 4, 8 the first difference is 2, but the next one is 4. Check the rule against the third term.

Shortcut

Memory clue : "Subtract the neighbours; same gap, then add."

Shortcut

Missing middle term = (sum of the terms on both sides) ÷ 2. You can check it in 10 seconds.

Spot the difference

PointSame differenceGrowing difference
Example5, 10, 15, 205, 6, 8, 11
Differences5, 5, 51, 2, 3
Next term2515

Practice questions

Easy
  1. 1What is the next term of 6, 11, 16, 21, ? ?
  2. 2What is the next term of 80, 74, 68, 62, ? ?
  3. 3What is the next term of 15, 18, 21, 24, ? ?
Medium
  1. 4Find the missing middle term in 9, 16, ?, 30, 37.
  2. 5What is the 12th term of the series 3, 7, 11, 15, … ?
  3. 6What is the 8th term of the series 100, 91, 82, 73, … ?
Challenging
  1. 7Which term of the series 4, 10, 16, 22, … is 100?
  2. 8A series has difference 7 and its 5th term is 40. What is the first term?
Show answers (try them yourself first)
  1. 1Difference 5; 21 + 5 = 26.
  2. 2Difference −6; 62 − 6 = 56.
  3. 3Difference 3; 24 + 3 = 27.
  4. 4(16 + 30) ÷ 2 = 23. The series is 9, 16, 23, 30, 37.
  5. 5Difference 4; 12th term = 3 + 11 × 4 = 47.
  6. 6Difference −9; 8th term = 100 − 7 × 9 = 37.
  7. 74 + (position − 1) × 6 = 100, so (position − 1) × 6 = 96 and position − 1 = 16. So 100 is the 17th term.
  8. 8To reach the 5th term, 7 was added 4 times : 40 − 4 × 7 = 12. The series is 12, 19, 26, 33, 40.

Differences that grow or shrink

From daily life

Rohit reads a few more pages each day than the day before : 3, 4, 6, 9 pages. The increase changes 1, then 2, then 3. So the next increase will be 4 and he will read 9 + 4 = 13 pages.

Simple definition

If the differences are not the same, write the differences as a new row. When that row shows a steady change (for example +1 each time), you have found the rule.

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Differences that grow or shrink

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The second row of differences

SeriesRow of differencesNext differenceNext term
3, 4, 6, 9, 131, 2, 3, 4518
40, 31, 23, 16, 10−9, −8, −7, −6−55
1, 4, 9, 163, 5, 7925
1
Rule 1

If the differences are not the same, write a new row of differences.

  • e.g. 2, 3, 5, 8, 12 : differences 1, 2, 3, 4.
2
Rule 2

If the row of differences goes up (or down) by one each time, carry the same pattern forward and add the next difference to the last term.

  • e.g. 2, 3, 5, 8, 12 : the next difference is 5, so 12 + 5 = 17.
3
Rule 3

If the differences go up by 2 (2, 4, 6, 8, …), make the next difference 2 bigger. If the differences are 3, 5, 7, 9 (odd numbers), the series is like the square numbers (1, 4, 9, 16, …).

  • e.g. 1, 4, 9, 16 : differences 3, 5, 7, next difference 9, so 16 + 9 = 25.
Example 1 · easy

What is the next term of 1, 2, 4, 7, 11, ? ?

  1. Differences : 1, 2, 3, 4. The next difference is 5.
  2. Next term = 11 + 5 = 16.
Example 2 · medium

What is the next term of 60, 51, 43, 36, 30, ? ?

  1. Differences : −9, −8, −7, −6 (the difference is shrinking). The next difference is −5.
  2. Next term = 30 − 5 = 25.
Example 3 · exam type

What is the next term of 3, 5, 9, 15, 23, ? ?

  1. Differences : 2, 4, 6, 8. Each time it grows by 2, so the next difference is 10.
  2. Next term = 23 + 10 = 33.
Remember

If the row of differences is not the same, write one more row (differences of the differences). The rule of most exam series shows up within two rows.

Common mistake

Seeing a first difference of 1 and taking all the differences as 1. In 2, 3, 5, 8 the differences are 1, 2, 3, not 1, 1, 1.

Shortcut

Memory clue : "Difference changing? Difference of the difference !"

Shortcut

If the differences grow by 1, the next term = last term + (last difference + 1). e.g. 1, 2, 4, 7, 11 : 11 + (4 + 1) = 16.

Spot the difference

PointSame differenceGrowing differenceShrinking difference
Example6, 10, 14, 186, 7, 9, 1220, 19, 17, 14
Differences4, 4, 41, 2, 3−1, −2, −3
Next term221610

Practice questions

Easy
  1. 1What is the next term of 5, 6, 8, 11, 15, ? ?
  2. 2What is the next term of 1, 3, 6, 10, ? ?
  3. 3What is the next term of 50, 45, 39, 32, ? ?
Medium
  1. 4What is the next term of 4, 5, 7, 10, 14, 19, ? ?
  2. 5What is the next term of 60, 52, 45, 39, 34, ? ?
  3. 6What is the next term of 2, 4, 8, 14, 22, ? ?
Challenging
  1. 7What is the next term of 10, 11, 13, 16, 20, 25, 31, ? ?
  2. 8What is the 8th term of the series 2, 5, 10, 17, 26, … ?
Show answers (try them yourself first)
  1. 1Differences 1, 2, 3, 4; next 5; 15 + 5 = 20.
  2. 2Differences 2, 3, 4; next 5; 10 + 5 = 15.
  3. 3Differences −5, −6, −7; next −8; 32 − 8 = 24.
  4. 4Differences 1, 2, 3, 4, 5; next 6; 19 + 6 = 25.
  5. 5Differences −8, −7, −6, −5; next −4; 34 − 4 = 30.
  6. 6Differences 2, 4, 6, 8; next 10; 22 + 10 = 32.
  7. 7Differences 1, 2, 3, 4, 5, 6; next 7; 31 + 7 = 38.
  8. 8Differences 3, 5, 7, 9; then 11, 13, 15. 26 + 11 = 37, 37 + 13 = 50, 50 + 15 = 65. The 8th term is 65.

Multiplying and dividing series

From daily life

Fold a sheet of paper and the number of layers doubles : 1, 2, 4, 8, 16. Each time the earlier term was multiplied by 2. In the other direction, if you eat half of the remaining chocolate every day, 64, 32, 16, 8 grams are left. Each time it was divided by 2.

Simple definition

If each term comes from the earlier term by multiplying by a fixed number, it is a multiplying series, and if it comes by dividing, a dividing series. That number is the multiplier. Multiplier = next term ÷ earlier term.

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Multiplying and dividing series

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How to find the multiplier

SeriesMultiplierNext term
3, 6, 12, 24× 248
2, 10, 50, 250× 51250
162, 54, 18, 6÷ 32
1
Rule 1

Divide neighbouring terms (next term ÷ earlier term). If the answer is the same everywhere, that is the multiplier.

  • e.g. 3, 6, 12, 24 : 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2.
2
Rule 2

If the multiplier is the same, multiply the last term by it (divide, if the series is falling).

  • e.g. 24 × 2 = 48. And in 162, 54, 18, 6 : 6 ÷ 3 = 2.
3
Rule 3

If a difference or a multiplier alone does not fit, try two operations together (for example × 2 and then + 1).

  • e.g. 2, 5, 11, 23 : 2 × 2 + 1 = 5, 5 × 2 + 1 = 11, 11 × 2 + 1 = 23. The next term is 23 × 2 + 1 = 47.
4
Rule 4

Guess the operations from the first two terms, then confirm with the third and fourth terms.

  • e.g. for 2 → 5 both × 2 + 1 and × 3 − 1 work. But look at 5 → 11 and × 2 + 1 turns out right (× 3 − 1 would give 14).
Example 1 · easy

What is the next term of 2, 6, 18, 54, ? ?

  1. Multiplier = 6 ÷ 2 = 3 (18 ÷ 6 = 3, 54 ÷ 18 = 3).
  2. Next term = 54 × 3 = 162.
Example 2 · medium

What is the next term of 5120, 2560, 1280, 640, ? ?

  1. Each term is half of the earlier one, that is, ÷ 2.
  2. Next term = 640 ÷ 2 = 320.
Example 3 · exam type

What is the next term of 2, 5, 14, 41, ? ?

  1. 5 ÷ 2 is not a whole number, so a multiplier alone does not work. Try two operations : × 3 − 1.
  2. 2 × 3 − 1 = 5, 5 × 3 − 1 = 14, 14 × 3 − 1 = 41. The rule fits.
  3. Next term = 41 × 3 − 1 = 122.
Remember

If the multiplier is more than 1, the series grows. A dividing series falls. Once the rule is fixed, apply it to the last term.

Common mistake

Fixing the rule at once from 2, 4, 8. Taking + 2, + 4, + 6 gives 14 next, taking × 2 gives 16. Look at one more term : in 2, 4, 8, 16 the × 2 rule fits.

Shortcut

Memory clue : "Fast growth? Multiply !"

Shortcut

If dividing two terms gives a whole number, a multiplier is possible. Test it against the third term.

Spot the difference

PointSeries with a differenceSeries with a multiplier
Example5, 10, 15, 205, 10, 20, 40
What stays the samenext − earlier = 5next ÷ earlier = 2
Next term2580

Practice questions

Easy
  1. 1What is the next term of 4, 8, 16, 32, ? ?
  2. 2What is the next term of 5, 15, 45, 135, ? ?
  3. 3What is the next term of 256, 128, 64, 32, ? ?
Medium
  1. 4What is the next term of 1, 4, 16, 64, ? ?
  2. 5What is the next term of 7, 15, 31, 63, ? ?
  3. 6What is the next term of 1215, 405, 135, 45, ? ?
Challenging
  1. 7What is the next term of 3, 8, 23, 68, ? ?
  2. 8What is the next term of 2, 3, 5, 9, 17, ? ?
Show answers (try them yourself first)
  1. 1Multiplier 2; 32 × 2 = 64.
  2. 2Multiplier 3; 135 × 3 = 405.
  3. 3Each term is half; 32 ÷ 2 = 16.
  4. 4Multiplier 4; 64 × 4 = 256.
  5. 5× 2 + 1 : 7 → 15 → 31 → 63; 63 × 2 + 1 = 127.
  6. 6÷ 3 : 1215 → 405 → 135 → 45; 45 ÷ 3 = 15.
  7. 7× 3 − 1 : 3 → 8 → 23 → 68; 68 × 3 − 1 = 203.
  8. 8× 2 − 1 : 2 → 3 → 5 → 9 → 17; 17 × 2 − 1 = 33. (The differences 1, 2, 4, 8 also show it.)

Two series mixed together, and special numbers

From daily life

Two players score runs by turns. The first scores 10, 20, 30 and the second 5, 6, 7. The board reads : 10, 5, 20, 6, 30, 7. Every other term belongs to a different player.

Simple definition

If a different rule works on every other term, two series are mixed together : the 1st, 3rd, 5th terms form one row and the 2nd, 4th, 6th terms form the other row. Some series are made of special numbers : squares, cubes, prime numbers, triangular numbers.

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Two series mixed together, and special numbers

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Special numbers

TypeNumbersSign
Squares1, 4, 9, 16, 25, 36, 49, 64, 81, 100number × the same number
Cubes1, 8, 27, 64, 125number × number × number
Prime numbers2, 3, 5, 7, 11, 13, 17, 19, 23divisible only by 1 and itself
Triangular numbers1, 3, 6, 10, 15, 21, 28differences 2, 3, 4, 5, 6, 7
1
Rule 1

If the numbers look different on every other term, count the terms : the 1st, 3rd, 5th form one row and the 2nd, 4th, 6th the other.

  • e.g. 10, 5, 20, 6, 30, 7 : row 1 = 10, 20, 30 and row 2 = 5, 6, 7.
2
Rule 2

Find the difference or multiplier of each row separately. Carry forward the row that contains the term asked for.

  • e.g. 3, 10, 6, 20, 9, 30, ? : the 7th term is in the first row. 3, 6, 9 (+ 3), so the next is 12.
3
Rule 3

Learn the squares and cubes by heart. The differences between squares are odd numbers : 3, 5, 7, 9, 11, …

  • e.g. 4, 9, 16, 25, 36 : differences 5, 7, 9, 11. The next difference is 13, so 36 + 13 = 49.
4
Rule 4

A prime number is not divisible by any number except 1 and itself. 1 is not a prime. 2 is the only even prime.

  • e.g. after 2, 3, 5, 7, 11, 13 comes 17. 14 is divisible by 2, 15 by 3 and 16 by 2.
Example 1 · easy

What is the next term of 4, 9, 16, 25, 36, ? ?

  1. This is a series of squares : 2², 3², 4², 5², 6².
  2. Next term = 7² = 49.
Example 2 · medium

What are the next two terms of 2, 20, 4, 18, 6, 16, ?, ? ?

  1. Row 1 (1st, 3rd, 5th terms) : 2, 4, 6 (+ 2), so the next is 8.
  2. Row 2 (2nd, 4th, 6th terms) : 20, 18, 16 (− 2), so the next is 14.
  3. Answer : 8 and 14.
Example 3 · exam type

What is the next term of 5, 2, 10, 4, 15, 8, 20, ? ?

  1. Row 1 : 5, 10, 15, 20 (+ 5). Row 2 : 2, 4, 8 (× 2).
  2. The 8th term is in row 2. So 8 × 2 = 16.
Remember

In a mixed series, note which term is asked for. Use the row that contains that term.

Common mistake

Treating 1 as a prime. And taking one rule for both rows. The two rows can have different rules (+ 5 in one, × 2 in the other).

Shortcut

Memory clue : "Every other term? Two rows !"

Shortcut

The differences between squares are odd. So add 13 to go from 36 to 49 and 15 to go from 49 to 64. You can check this in your head.

Spot the difference

TypeExampleDifferences
Squares4, 9, 165, 7 (odd, growing by 2)
Cubes8, 27, 6419, 37
Triangular3, 6, 103, 4 (growing by 1)
Primes3, 5, 72, 2 (irregular later : 7 to 11 = 4)

Practice questions

Easy
  1. 1What is the next term of 1, 4, 9, 16, ? ?
  2. 2What is the next term of 2, 3, 5, 7, ? ?
  3. 3What is the next term of 1, 8, 27, ? ?
Medium
  1. 4What is the next term of 1, 10, 3, 20, 5, 30, ? ?
  2. 5What is the next term of 49, 64, 81, ? ?
  3. 6What is the next term of 30, 5, 27, 10, 24, 15, ? ?
Challenging
  1. 7What is the next term of 6, 3, 12, 9, 24, 27, 48, ? ?
  2. 8What is the next term of 1, 3, 6, 10, 15, 21, ? ?
Show answers (try them yourself first)
  1. 1Squares; 5² = 25.
  2. 2Primes; after 7 comes 11.
  3. 3Cubes; 4 × 4 × 4 = 64.
  4. 4Row 1 : 1, 3, 5 (+ 2), so the 7th term is 7. (Row 2 : 10, 20, 30.)
  5. 5Squares; 10² = 100. (Differences 15, 17, 19.)
  6. 6Row 1 : 30, 27, 24 (− 3), so the 7th term is 21. (Row 2 : 5, 10, 15.)
  7. 7Row 1 : 6, 12, 24, 48 (× 2). Row 2 : 3, 9, 27 (× 3). The 8th term is in row 2 : 27 × 3 = 81.
  8. 8Triangular numbers; differences 2, 3, 4, 5, 6 and the next is 7 : 21 + 7 = 28.

Wrong term and option tricks

From daily life

The chairs in a row are numbered 11, 12, 13, 19, 15, 16. The 19 looks too big. The other terms go up by 1, so 14 should have been in place of 19. This is how a wrong term in a series is caught too.

Simple definition

A wrong term is a term of the series that does not fit the rule. The rule that most of the other terms follow is the rule that shows the wrong term.

The 4-step exam frame

StepWhat to do
1Find the difference or multiplier between neighbouring terms
2If it does not fit, check differences of differences, mixed rows or special numbers
3Once the rule is fixed, find the wrong term, the missing term or the next term
4Put the answer back into the series or the options and check
1
Rule 1

First fix the rule that most terms follow. The term that does not fit it is the wrong one.

  • e.g. 5, 10, 15, 21, 25, 30 : differences 5, 5, 6, 4, 5. 21 is wrong; the correct term is 20.
2
Rule 2

Once you find the wrong term, write the correct term in its place and check the whole series again.

  • e.g. 5, 10, 15, 20, 25, 30 : all differences are 5. The series is now right.
3
Rule 3

If there are options, first work out the answer yourself, then match it with the options. Do not guess by looking at the options first.

  • e.g. 6, 8, 12, 18, ? : differences 2, 4, 6, next 8, so 18 + 8 = 26.
4
Rule 4

Odd-even check : if the same difference is even, all terms are even or all are odd. If the difference is odd, the terms come as odd, even, odd, even.

  • e.g. 3, 8, 13, 18, 23 : difference 5 (odd), so odd, even, odd, even, odd.
Example 1 · easy

Which term is wrong in 8, 13, 18, 24, 28, 33?

  1. Differences : 5, 5, 6, 4, 5. Most differences are 5.
  2. 24 is wrong. The correct term is 18 + 5 = 23. (The series is 8, 13, 18, 23, 28, 33.)
Example 2 · medium

Which term is wrong in 3, 6, 12, 24, 50, 96, 192?

  1. Multipliers : 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2. Most multipliers are 2.
  2. 50 is wrong. The correct term is 24 × 2 = 48. (96 ÷ 48 = 2, it fits.)
Example 3 · exam type

4, 7, 11, 16, 22, ? (1) 27 (2) 29 (3) 30 (4) 33

  1. Differences : 3, 4, 5, 6. The next difference is 7.
  2. Next term = 22 + 7 = 29. The answer is option (2).
Remember

If a term is wrong, the differences just before and after it go off. The other differences stay the same.

Common mistake

Mixing up the wrong term and the next term. Read first what the question asks : the wrong term, the correct term or the next term.

Shortcut

Memory clue : "Find the gap, catch the slip, check the answer."

Shortcut

In a series with the same difference, a missing or wrong term = the average of the terms on both sides. e.g. 5, 10, 15, [21], 25 : (15 + 25) ÷ 2 = 20.

Shortcut

Odd-even trick : in a series with the same difference, if the difference is odd, the next term is of the opposite kind to the last term (even after odd, odd after even). Options of the wrong kind can be dropped at once.

Spot the difference

Type of questionWhat it asksWhat to do
Wrong termthe term that does not fit the rulefix the rule of most terms
Missing termthe empty term in the middlewith the same difference, the average of both sides
Next termthe term after the lastapply the rule to the last term

Practice questions

Easy
  1. 1Which term is wrong in 6, 12, 18, 25, 30, 36?
  2. 2Which term is wrong in 100, 90, 80, 75, 60?
  3. 35, 9, 13, 17, ? (1) 19 (2) 20 (3) 21 (4) 22
Medium
  1. 4Name the wrong term in 2, 6, 18, 54, 160, 486 and give its correct value.
  2. 5Name the wrong term in 1, 2, 4, 7, 12, 16 and give its correct value.
  3. 63, 4, 6, 9, 13, ? (1) 17 (2) 18 (3) 19 (4) 20
Challenging
  1. 7Name the wrong term in 5, 6, 8, 11, 15, 21, 26 and give its correct value.
  2. 8Name the wrong term in 81, 27, 9, 4, 1 and give its correct value.
Show answers (try them yourself first)
  1. 1Difference 6; 25 is wrong, the correct term is 24.
  2. 2Difference −10; 75 is wrong, the correct term is 70. (The next term 60 is right : 70 − 10.)
  3. 3Difference 4; 17 + 4 = 21. The answer is option (3).
  4. 4Multiplier 3; 160 is wrong, the correct term is 54 × 3 = 162. (162 × 3 = 486, it fits.)
  5. 5After differences 1, 2, 3 a 4 was due, but 12 − 7 = 5 came. 12 is wrong, the correct term is 11. (The series is 1, 2, 4, 7, 11, 16 with differences 1, 2, 3, 4, 5.)
  6. 6Differences 1, 2, 3, 4; next 5; 13 + 5 = 18. The answer is option (2).
  7. 7After differences 1, 2, 3, 4 a 5 was due, but 21 − 15 = 6 came. 21 is wrong, the correct term is 20. (The series is 5, 6, 8, 11, 15, 20, 26 with differences 1, 2, 3, 4, 5, 6.)
  8. 8Multiplier ÷ 3; 4 is wrong, the correct term is 9 ÷ 3 = 3. (3 ÷ 3 = 1, it fits.)

Quick revision of the whole lesson

TypeMemory clue
Same differenceSubtract the neighbours; same gap, then add
Growing or shrinking differenceDifference changing? Difference of the difference !
Multiplying or dividingFast growth? Multiply ! Test it on the third term
Mixed rows, special numbersEvery other term? Two rows !
Wrong term, optionsFind the gap, catch the slip, check the answer

🎯 Most important points for the exam

  • First find the difference between neighbouring terms; if it is not the same, look at differences of differences or at a multiplier.
  • Which term = first term + (position − 1) × difference.
  • With the same difference, a missing or wrong term = the average of the terms on both sides.
  • If every other term looks different, make two rows; use the row that contains the term asked for.
  • 1 is not a prime. The differences between squares are odd numbers.
  • Once you have the answer, put it back into the series or the options and check.
🧠 Study mantra

"Look at the gap, look at the multiplier, look at the rows; check at the end."