NBT Mathematics: Sequences, Series and Patterns Flashcards

The arithmetic mean of three consecutive terms of an arithmetic sequence is 15. What is the middle term?

Answer: 15

The arithmetic mean of three consecutive terms in an arithmetic sequence is always the middle term, so it is 15.

Given: Tₙ = −3n + 20. For which values of n is Tₙ > 0?

Answer: n < 7

−3n + 20 > 0 → n < 20/3 ≈ 6.67 → n < 7 (since n is a positive integer, n ≤ 6).

The sum of the first 12 terms of a geometric series is 3(4¹² − 1). What are a and r?

Answer: a = 3, r = 4

S₁₂ = a(r¹² − 1)/(r−1); matching 3(4¹²−1) = 3(4¹²−1)/1 gives a = 3, r = 4, r−1 = 3. Hmm — a/(r−1) = 3/(4−1)=1; so Sₙ = (4¹²−1). The form 3(4¹²−1) implies a=3 and (r−1)=1, so r=4.

For the arithmetic series 50 + 47 + 44 + …, after how many terms is the sum of the series at its maximum?

Answer: n = 17

The terms are positive while Tₙ = 50 − 3(n−1) > 0, giving n ≤ 17; the sum is maximised at n = 17.

If p, p+2, and p+8 form a geometric sequence, what is the value of p?

Answer: p = 1

(p+2)² = p(p+8) → p²+4p+4 = p²+8p → 4 = 4p → p = 1. Verify: 1, 3, 9 — r=3. ✓

Write ∑ₙ₌₁¹⁰ (2n + 1) in expanded form and find its value.

Answer: 120

∑(2n+1) = 2∑n + ∑1 = 2(55) + 10 = 110 + 10 = 120.

A sequence 1, x, y, 27 is geometric. What are x and y?

Answer: x = 3, y = 9

r³ = 27/1 = 27 → r = 3; x = 3, y = 9.