Arithmetic progressions
Class 10 Maths · Arithmetic progressions · 26 questions with answers
Arithmetic progressions is the most reliably scored chapter in the Class 10 paper: two formulas, a fixed method, and questions that repeat year after year. It is also where careless marks go - a common difference read as the second term, or n found and then not used. Both are habits rather than misunderstandings.
- Identifies the first term and the common difference correctly
- Finds the nth term, and finds which term a given value is
- Finds the sum of the first n terms
- Solves for a and d from two given terms
- Applies an AP to a described situation
Exercise 1 of 3 · 10 questions
Find what is asked.
Write the answer · Warm-up
Here’s one done for you
The last question is the useful one. If solving for n does not give a positive whole number, the value is not a term of the AP - and saying that explicitly is the answer rather than a dead end. Children often assume they have made an arithmetic mistake and search for one. The non-integer is the result.
- 1)Find the 10th term of 2, 7, 12, ...
- 2)Which term of 3, 8, 13, ... is 78?
- 3)Find the common difference of 10, 7, 4, ...
- 4)Find the 15th term of the AP whose first term is -5 and common difference is 3.
- 5)Is 100 a term of the AP 3, 7, 11, ...?
- 6)Find the 12th term of 5, 9, 13, ...
- 7)Which term of 7, 13, 19, ... is 103?
- 8)Find the first term of an AP whose 5th term is 20 and common difference is 3.
- 9)Is 68 a term of the AP 4, 9, 14, ...?
- 10)Find the common difference of an AP whose 3rd term is 11 and 8th term is 31.
Answers
- 1) a = 2, d = 5, so a₁₀ = 2 + 9(5) = 47.
- 2) 3 + (n - 1)5 = 78 gives (n - 1)5 = 75 and n = 16. It is the 16th term.
- 3) d = 7 - 10 = -3.
- 4) a₁₅ = -5 + 14(3) = 37.
- 5) 3 + (n - 1)4 = 100 gives (n - 1)4 = 97, which is not a whole number when divided by 4. So 100 is not a term of this AP.
- 6) a = 5, d = 4, so a₁₂ = 5 + 11(4) = 49.
- 7) a = 7, d = 6, so 7 + (n - 1)6 = 103 gives n - 1 = 16 and n = 17.
- 8) a + 4(3) = 20, so a = 8.
- 9) 4 + (n - 1)5 = 68 gives (n - 1)5 = 64, which is not a whole number, so 68 is not a term.
- 10) a + 7d - (a + 2d) = 31 - 11, so 5d = 20 and d = 4.
Exercise 2 of 3 · 8 questions
Find the sum.
Write the answer · Guided
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When the last term is known, use S = (n/2)(a + l) rather than the longer formula - it is faster and there is less to get wrong. The step people skip is finding n first, using n = (l - a)/d + 1. That '+1' is the commonest slip in the chapter: from 12 to 99 in steps of 3 there are 29 gaps and 30 numbers.
- 1)The sum of the first 20 terms of 1, 3, 5, ...
- 2)The sum of the first 15 terms of the AP with a = 3 and d = 4.
- 3)How many two-digit numbers are divisible by 3?
- 4)Find 7 + 10.5 + 14 + ... + 84.
- 5)Find the sum of the first 25 terms of 3, 7, 11, ...
- 6)Find the sum of the first 12 terms of 2, 5, 8, ...
- 7)How many two-digit numbers are divisible by 4, and what is their sum?
- 8)Find the sum of all multiples of 7 lying between 100 and 200.
Answers
- 1) S₂₀ = (20/2)[2(1) + 19(2)] = 10(40) = 400.
- 2) S₁₅ = (15/2)[2(3) + 14(4)] = (15/2)(62) = 465.
- 3) They form the AP 12, 15, ..., 99. n = (99 - 12)/3 + 1 = 30. There are 30 such numbers.
- 4) a = 7, d = 3.5, last term 84. n = (84 - 7)/3.5 + 1 = 23. S = (23/2)(7 + 84) = (23/2)(91) = 1046.5.
- 5) S₂₅ = 25/2 [2(3) + 24(4)] = 25/2 × 102 = 1,275.
- 6) S₁₂ = 12/2 [2(2) + 11(3)] = 6 × 37 = 222.
- 7) They run 12, 16, ..., 96, so n = 22. Sum = 22/2 (12 + 96) = 11 × 108 = 1,188.
- 8) They run 105, 112, ..., 196, so n = 14. Sum = 14/2 (105 + 196) = 7 × 301 = 2,107.
Exercise 3 of 3 · 8 questions
Work these out.
Write the answer · Practice
Here’s one done for you
The second question is the one worth learning properly, because it looks unfamiliar and has a one-line method: the nth term is the sum of n terms minus the sum of (n-1) terms. Here that is S₁₀ - S₉ = 350 - 288 = 62. Any question giving you Sₙ as a formula is asking you to use that relationship, and recognising it turns a hard-looking question into an easy one.
- 1)The 5th term of an AP is 19 and the 9th is 35. Find the first term and the common difference.
- 2)The sum of the first n terms of an AP is 3n² + 5n. Find its 10th term.
- 3)A person saves ₹100 in the first month and ₹50 more each month than the month before. How much will they have saved in a year?
- 4)Find the number of terms in the AP 18, 15.5, 13, ..., -47.
- 5)The 4th term of an AP is 14 and the 10th is 32. Find the first term and the common difference.
- 6)The sum of the first 10 terms of an AP is 155 and the first term is 2. Find the common difference.
- 7)Find the middle term of the AP 6, 13, 20, ..., 216.
- 8)Which term of the AP 5, 2, -1, ... is -49?
Answers
- 1) a + 4d = 19 and a + 8d = 35. Subtracting gives 4d = 16, so d = 4 and a = 3.
- 2) a₁₀ = S₁₀ - S₉ = (300 + 50) - (243 + 45) = 350 - 288 = 62.
- 3) a = 100, d = 50, n = 12. S = (12/2)[200 + 11(50)] = 6(750) = ₹4,500.
- 4) d = -2.5. Then 18 + (n - 1)(-2.5) = -47 gives (n - 1)(-2.5) = -65, so n - 1 = 26 and n = 27.
- 5) a + 3d = 14 and a + 9d = 32, so 6d = 18, d = 3 and a = 5.
- 6) 155 = 10/2 [2(2) + 9d] = 5(4 + 9d), so 31 = 4 + 9d and d = 3.
- 7) a = 6, d = 7, and 6 + (n - 1)7 = 216 gives n = 31. The middle term is the 16th: 6 + 15(7) = 111.
- 8) a = 5, d = -3, and 5 + (n - 1)(-3) = -49 gives (n - 1)(-3) = -54, so n = 19.
While your child works
- Check the common difference is second minus first, not the second term itself. That single slip ruins otherwise perfect work.
- n = (l - a)/d + 1. The '+1' is the commonest error in the chapter - 29 gaps means 30 terms.
- If solving for n gives a fraction, the value is not a term of the AP. That is the answer, not a mistake.