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CBSE · Class 10 · Mathematics

Arithmetic Progressions 5.3: nth Term of an AP

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

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Finding a term using aₙ = a + (n − 1)d

From daily life

To reach the 10th step of a staircase you do not climb every step: you start at a and add d nine times.

Simple definition

The nth term of an AP with first term a and common difference d is aₙ = a + (n − 1)d. The (n − 1) is the number of steps taken from the first term.

Formula: aₙ = a + (n − 1)d

astart+ (n − 1)steps× deach step= aₙ

In the figure: 3, 7, 11, … : a = 3, d = 4, so a₈ = 3 + 7 × 4 = 31.

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Finding a term using aₙ = a + (n − 1)d

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Rule 1: Find the term — Find a and d, then substitute n.

  • Easy: a = 5, d = 3: a₁₀ = 5 + 27 = 32.
  • Medium: 20, 17, 14, … : a₁₂ = 20 + 11(−3) = −13.
  • Exam-type: −10, −7, −4, … : a₉ = −10 + 8 × 3 = 14.

Rule 2: Find the last term — The last term of an n-term AP is aₙ.

  • Easy: AP of 20 terms with a = 2, d = 3: l = 2 + 19 × 3 = 59.
  • Medium: Second term a₂ = a + d.
  • Exam-type: First term a₁ = a + 0 × d = a.
Remember

The nth term uses (n − 1) steps, not n steps.

Common mistake

Using n × d instead of (n − 1) × d.

Memory clue

"One fewer step than the position."

Practice questions

Easy
  1. 1The nth term of an AP is aₙ = 5n + 10. Find the 25th term.
  2. 2The nth term of an AP is aₙ = −7n − 2. Find the common difference.
  3. 3An AP has first term 12 and common difference −5. What is its 28th term?
Medium
  1. 4The nth term of an AP is aₙ = −n + 11. Find the first term.
  2. 5The nth term of an AP is aₙ = 6n + 13. Find the common difference.
  3. 6An AP has first term 21 and common difference 8. What is its 43rd term?
Challenging
  1. 7An AP has first term 1 and common difference −5. What is its 34th term?
  2. 8The nth term of an AP is aₙ = 5n + 13. Find the common difference.
Show answers (try them yourself first)
  1. 1135
  2. 2−7
  3. 3−123
  4. 410
  5. 56
  6. 6357
  7. 7−164
  8. 85

Finding the position of a term

From daily life

You see 47 in the list and wonder: which step is that? Solve aₙ = 47 for n.

Simple definition

To find the position n of a given term T: put a + (n − 1)d = T and solve, n = (T − a)/d + 1. If n is not a positive whole number, T is not a term.

Formula: n = (T − a)/d + 1

47 = 3 + (n−1)·4(n−1) = 11n = 12

In the figure: In 3, 7, 11, …, the term 47 is the 12th: 47 = 3 + (n − 1)4 gives n = 12.

Rule 1: Which term is it? — Solve for n and check it is a positive whole number.

  • Easy: 3, 7, 11, … : 47 is the 12th term.
  • Medium: 100, 95, 90, … : 0 is the 21st term (n = 100/5 + 1).
  • Exam-type: 2, 5, 8, … : which term is 62? n = 60/3 + 1 = 21.

Rule 2: Is it a term? — If n is a fraction or negative, the number is not in the list.

  • Easy: Is 100 a term of 2, 5, 8, …? n = 98/3 + 1 not a whole number: no.
  • Medium: Is 77 a term of 2, 5, 8, …? n = 75/3 + 1 = 26: yes.
  • Exam-type: −5 is not a term of 3, 7, 11, …: n would be negative.
Remember

Check by putting n back: aₙ must equal T.

Common mistake

Forgetting to add 1 at the end: n = (T − a)/d + 1.

Memory clue

"Divide the gap by d, then add one."

Practice questions

Easy
  1. 1Is 134 a term of the AP 15, 22, 29, 36, …?
  2. 2Which term of the AP 3, −2, −7, −12, … is −77?
  3. 3Is 94 a term of the AP 3, 10, 17, 24, …?
Medium
  1. 4Is 51 a term of the AP −6, −2, 2, 6, …?
  2. 5Is 157 a term of the AP 19, 27, 35, 43, …?
  3. 6Which term of the AP 5, 12, 19, 26, … is 117?
Challenging
  1. 7Which term of the AP 18, 15, 12, 9, … is −9?
  2. 8Which is the first term of the AP 20, 23, 26, 29, … that is greater than 330?
Show answers (try them yourself first)
  1. 1Yes
  2. 217
  3. 3Yes
  4. 4No
  5. 5No
  6. 617
  7. 710
  8. 8105

Finding a and d from given terms

From daily life

If you know the 3rd and 7th steps, the climb between them shows the size of one step.

Simple definition

Write each given term as a + (n − 1)d to get two equations. Subtract to find d, then substitute to find a.

a₃ = a + 2d = 12a₇ = a + 6d = 24subtract4d = 12d = 3, a = 6

In the figure: a₃ = 12 and a₇ = 24: 4d = 12, d = 3, a = 12 − 6 = 6.

Rule 1: Two terms given — Subtract to remove a.

  • Easy: a₃ = 12, a₇ = 24: d = 3, a = 6 (check a₇ = 6 + 18 = 24 ✓).
  • Medium: a₂ = 13, a₄ = 3: 2d = −10, d = −5, a = 18.
  • Exam-type: a₅ = 22 and d = 4: a = 22 − 16 = 6.

Rule 2: Then find any term — With a and d known, use aₙ = a + (n − 1)d.

  • Easy: a = 6, d = 3: a₁₅ = 6 + 14 × 3 = 48.
  • Medium: a = 18, d = −5: a₁₀ = 18 − 45 = −27.
  • Exam-type: Check both given terms with your a and d.
Remember

The gap between aₚ and aq is (q − p)d.

Common mistake

Dividing by the wrong number of steps (using 7 − 3 + 1 instead of 7 − 3).

Memory clue

"Subtract the positions to count the steps."

Practice questions

Easy
  1. 1In an AP, the first term is 15 and the 6th term is 35. Find the common difference.
  2. 2In an AP, the first term is 14 and the 6th term is 24. Find the common difference.
  3. 3In an AP, the first term is −6 and the 4th term is 15. Find the common difference.
Medium
  1. 4In an AP, the first term is 24 and the 4th term is 21. Find the common difference.
  2. 5In an AP, a13 − a5 = −72. Find the common difference.
  3. 6For an AP, the 5th term is −26 and the 4th term is −18. Find the first term.
Challenging
  1. 7In an AP, a2 + a2 = 0 and a3 + a8 = −28. Find the first term.
  2. 8For an AP, the 4th term is 42 and the 10th term is 90. Find the first term.
Show answers (try them yourself first)
  1. 14
  2. 22
  3. 37
  4. 4−1
  5. 5−9
  6. 66
  7. 74
  8. 818

Number of terms in an AP

From daily life

How many seats are in a row that starts at 7 and ends at 205 with steps of 6? It is the same as asking which term 205 is.

Simple definition

For a finite AP with first term a, last term l and difference d, the number of terms is n = (l − a)/d + 1.

Formula: n = (l − a)/d + 1

l − a÷ d+ 1= n

In the figure: 7, 13, 19, …, 205: n = (205 − 7)/6 + 1 = 34 terms.

Rule 1: Count the terms — Find d first, then use the formula.

  • Easy: 7, 13, 19, …, 205: 34 terms.
  • Medium: 2, 4, 6, …, 100: n = 98/2 + 1 = 50.
  • Exam-type: 50, 45, …, 5: n = (5 − 50)/(−5) + 1 = 10.

Rule 2: Counting numbers in a range — Find the first and last multiple in the range.

  • Easy: Two-digit multiples of 3: 12, 15, …, 99: n = 87/3 + 1 = 30.
  • Medium: Multiples of 5 from 10 to 95: n = 85/5 + 1 = 18.
  • Exam-type: n must be a whole number.
Remember

Counting terms is like counting fence posts: the gaps are one fewer than the posts.

Common mistake

Forgetting the +1, which counts the first term too.

Memory clue

"Gaps plus one equals terms."

Practice questions

Easy
  1. 1How many terms does the AP 5, 11, 17, …, 53 have?
  2. 2How many terms does the AP 13, 16, 19, …, 31 have?
  3. 3How many terms does the AP 12, 19, 26, …, 54 have?
Medium
  1. 4How many terms are there in the AP −15, −10, −5, …, 205?
  2. 5How many numbers from 8 to 364 (both included) are divisible by 13?
  3. 6How many terms are there in the AP 17, 12, 7, …, −158?
Challenging
  1. 7How many terms are there in the AP 8, 4, 0, …, −96?
  2. 8How many three-digit numbers are divisible by 11?
Show answers (try them yourself first)
  1. 19
  2. 27
  3. 37
  4. 445
  5. 528
  6. 636
  7. 727
  8. 881

Word problems on the nth term

From daily life

Savings, salaries and seat rows follow an AP. The nth term tells the amount on the nth week, year or row.

Simple definition

Identify a (the first value) and d (the fixed change), decide whether you need aₙ or n, solve, and write the answer with units.

finda and dchooseaₙ or nsolveanswerwith units

In the figure: Savings start at ₹50 and rise ₹10 each week: week 12 is 50 + 11 × 10 = ₹160.

Rule 1: Find the nth value — Use aₙ = a + (n − 1)d.

  • Easy: Salary ₹8000, rises ₹500 yearly: 10th year ₹8000 + 9 × 500 = ₹12500.
  • Medium: Seats: first row 20, each row 2 more: row 15 has 20 + 14 × 2 = 48.
  • Exam-type: Taxi: ₹15 then ₹8 per extra km: 10 km costs 15 + 9 × 8 = ₹87.

Rule 2: Find n — Equate the nth value to the given amount and solve for n.

  • Easy: Salary reaches ₹10000 in year n = 2000/500 + 1 = 5.
  • Medium: Savings ₹50, +10: reaches ₹200 in week 16.
  • Exam-type: Check n is a whole number before answering.
Remember

The units of aₙ are the units of the quantity, such as ₹ or seats.

Common mistake

Treating the first value as the zeroth step: the first value is a₁ = a.

Memory clue

"a is week one; weeks passed are n minus one."

Practice questions

Easy
  1. 1A stack of bricks has 9 bricks in the top layer and every layer below has 2 more than the layer above. How many bricks are in layer number 6 from the top?
  2. 2A stack of bricks has 3 bricks in the top layer and every layer below has 4 more than the layer above. How many bricks are in layer number 8 from the top?
  3. 3A stack of bricks has 10 bricks in the top layer and every layer below has 2 more than the layer above. How many bricks are in layer number 8 from the top?
Medium
  1. 4In an auditorium the first row has 15 seats and each next row has 5 seats more than the previous row. How many seats are there in row number 19?
  2. 5In an auditorium the first row has 17 seats and each next row has 4 seats more than the previous row. How many seats are there in row number 9?
  3. 6A person starts with a monthly salary of ₹25000 and gets an increment of ₹750 every year. What is the monthly salary in year 17 (in ₹)?
Challenging
  1. 7An AP has 21 terms, first term 7 and last term 67. Find its middle term.
  2. 8A ladder's first rung is 6 cm wide and each higher rung is 2 cm wider than the one below. How wide is the 19th rung (in cm)?
Show answers (try them yourself first)
  1. 119
  2. 231
  3. 324
  4. 4105
  5. 549
  6. 637000
  7. 737
  8. 842