NBT Mathematics: Sequences, Series and Patterns Flashcards

For a geometric sequence, T₃ = 12 and T₇ = 192. What is T₅?

Answer: 48

T₇/T₃ = r⁴ = 16 → r² = 4; T₅ = T₃ × r² = 12 × 4 = 48.

The sum of the first n terms of an arithmetic series is Sₙ = 2n² + 3n. What is T₅?

Answer: 21

T₅ = S₅ − S₄ = (2×25+15) − (2×16+12) = 65 − 44 = 21.

What is the general term of the sequence −2, 1, 4, 7, 10, …?

Answer: Tₙ = 3n − 5

a = −2, d = 3; Tₙ = −2 + (n−1)(3) = 3n − 5.

Evaluate: ∑ₖ₌₂⁵ (k² − 1)

Answer: 50

k=2: 3; k=3: 8; k=4: 15; k=5: 24. Sum = 3 + 8 + 15 + 24 = 50.

A sequence is defined by Tₙ = Tₙ₋₁ × 2 with T₁ = 3. What is the sum of the first 5 terms?

Answer: 93

Terms: 3, 6, 12, 24, 48. Sum = 3(2⁵ − 1)/(2−1) = 3(31) = 93.

An arithmetic sequence has a₁ = p and d = q. Express the sum of the first n terms in terms of p, q, n.

Answer: n/2 (2p + (n−1)q)

Sₙ = n/2 × [2a + (n−1)d] = n/2 × [2p + (n−1)q].

Which term of the geometric sequence 2, 6, 18, 54, … first exceeds 1000?

Answer: 7th

Tₙ = 2 × 3ⁿ⁻¹. T₆=486, T₇=1458 > 1000, so the 7th term is the first to exceed 1000.