NBT Mathematics: Sequences, Series and Patterns Flashcards

The 7th term of an arithmetic sequence is 43 and the common difference is 6. What is the first term?

Answer: 7

T₇ = a + 6d = a + 36 = 43 → a = 7.

For which value(s) of n does the series ∑ₖ₌₁ⁿ k = 55?

Answer: n = 10

Sₙ = n(n+1)/2 = 55 → n(n+1) = 110 → n = 10.

A pattern of dots: Row 1 has 1 dot, Row 2 has 3, Row 3 has 5. How many dots are in Row n?

Answer: 2n − 1

Each row adds 2 more dots; the nth row has 2n − 1 dots.

A geometric series has S∞ = 12 and r = 1/3. What is the first term?

Answer: 8

S∞ = a/(1−r) = a/(2/3) = 3a/2 = 12 → a = 8.

The second differences of a sequence are all equal to 6. If T₁ = 2 and T₂ = 5, what is T₃?

Answer: 14

First difference Δ₁ = T₂ − T₁ = 3; next first difference = 3 + 6 = 9; T₃ = 5 + 9 = 14.

Which sequence is neither arithmetic nor geometric?

Answer: 1, 2, 4, 7, 11

1, 2, 4, 7, 11 has increasing first differences (1, 2, 3, 4) so it is quadratic, not arithmetic or geometric.

Simplify: ∑ₖ₌₁⁶ (3k + 2)

Answer: 75

Sum = 3∑k + 2(6) = 3(21) + 12 = 63 + 12 = 75. Wait: 5+8+11+14+17+20 = 75.