NBT Mathematics: Sequences, Series and Patterns Flashcards

The sum of an arithmetic series is given by Sₙ = n² + 2n. What is the 6th term?

Answer: 13

T₆ = S₆ − S₅ = (36+12) − (25+10) = 48 − 35 = 13.

A ball is dropped from 16 m. Each bounce reaches 3/4 of the previous height. What is the total distance travelled when it hits the ground for the 4th time?

Answer: 52 m

Down+up pairs: 16 + 2(12) + 2(9) + 2(6.75) = 16 + 24 + 18 + 13.5 = 71.5 m — but only the fall portion by the 4th hit. Total = 16 + 12 + 9 + 6.75 + 12 + 9 + 6.75 = down 4 + up 3: 16+12+9+6.75 + 12+9+6.75 = 71.5 m. For just the 4th ground contact: total path = 16 + 2(12+9+6.75) = 16 + 2(27.75) = 71.5 m.

The 100th term of the arithmetic sequence with T₁ = 7 and d = −3 is:

Answer: −290

T₁₀₀ = 7 + (100−1)(−3) = 7 − 297 = −290.

Which expression equals ∑ₙ₌₁^∞ (1/3)ⁿ?

Answer: 1/2

S∞ = (1/3)/(1 − 1/3) = (1/3)/(2/3) = 1/2.

A recursive sequence is defined by Tₙ = 2Tₙ₋₁ − 3 with T₁ = 5. What is T₄?

Answer: 19

T₂=2(5)−3=7; T₃=2(7)−3=11; T₄=2(11)−3=19.

The first term of a geometric sequence is 5 and the sum of the first 4 terms is 75. What is the common ratio?

Answer: r = 2

5(r⁴−1)/(r−1) = 75 → (r⁴−1)/(r−1) = 15. For r=2: (16−1)/1=15 ✓.

In the pattern 1², 1²+2², 1²+2²+3², … the nth term is given by Tₙ = n(n+1)(2n+1)/6. What is T₄?

Answer: 30

T₄ = 4(5)(9)/6 = 180/6 = 30.