Find the nth term of each sequence.
(a) 8, 15, 34, 71, 132, …
(b) $\frac{2}{1}$, $\frac{3}{4}$, $\frac{4}{16}$, $\frac{5}{64}$, $\frac{6}{256}$, …
▶️ Answer/Explanation
(a) Ans: n³ + 7
Looking at differences: 7, 19, 37, 61 (second differences: 12, 18, 24). The cubic pattern suggests n³. Testing: 1³+7=8, 2³+7=15, etc.
(b) Ans: $\frac{n+1}{4^{n-1}}$
Numerator increases by 1: n+1. Denominator follows 4^(n-1) pattern: 4⁰=1, 4¹=4, 4²=16, etc.
(a) The nth term of a sequence is n² + 3n. Find the first three terms of this sequence.
(b) These are the first five terms of a different sequence.
25, 18, 11, 4, -3
Find the nth term of this sequence.
▶️ Answer/Explanation
(a) 4, 10, 18
Substitute n=1: 1² + 3×1 = 4
n=2: 2² + 3×2 = 10
n=3: 3² + 3×3 = 18
(b) 32 – 7n
Difference is -7 each time. First term is 25 when n=1: 25 = a + (-7×1) → a = 32