Sequences and Series — Practice

Question 1

Find the 20th term and sum of first 20 terms of the AP: 3, 7, 11, 15, ...

Accepted Answer

a = 3, d = 7−3 = 4. a₂₀ = 3+(20−1)×4 = 3+76 = 79. S₂₀ = 20/2×(3+79) = 10×82 = 820.

Question 2

The sum of infinite terms of a GP is 12 and its first term is 4. Find the common ratio and the 4th term.

Accepted Answer

S∞ = a/(1−r) → 12 = 4/(1−r) → 1−r = 1/3 → r = 2/3. Check: |2/3| < 1 ✓. a₄ = 4×(2/3)³ = 4×8/27 = 32/27.

Question 3

If a, b, c are in AP and a, b, c are all positive, prove that a + c ≥ 2b, i.e., AM ≥ GM gives the same inequality.

Accepted Answer

Since a, b, c are in AP: b − a = c − b, so 2b = a + c. This means b = (a+c)/2 = AM of a and c. By AM ≥ GM: (a+c)/2 ≥ √(ac), i.e., b ≥ √(ac). So the middle term of an AP is the AM of the two outer terms, and this is ≥ the geometric mean √(ac). Equality holds when a = c.

Practice — Sequences and Series

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