Sequences & Series • Topic 3 of 3

Recurrence Relations

A recurrence defines each term from the ones before it. The most famous is Fibonacci, F(n) = F(n−1) + F(n−2), starting 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... CAT rarely needs a closed form here; it needs you to compute a few terms carefully or to spot a useful identity. Two identities pay off: the sum of the first n Fibonacci numbers is F(n+2) − 1, and the running pattern of a recurrence often repeats or cycles — so when a problem asks for a far-out term, check whether the sequence is periodic (especially when terms are taken modulo something, or defined by alternating signs). For an AGP (arithmetic-geometric progression), where the nth term is an AP term times a GP term such as n·2^(n−1), the standard technique is the S − rS shift: write the sum S, multiply it by the common ratio r, subtract, and the staggered terms collapse into a plain GP plus boundary terms. For an infinite AGP with |r| < 1 the sum is a/(1−r) + dr/(1−r)². The exam-smart habit: for any recurrence, unroll three or four terms before reaching for theory — the pattern is often obvious.

✅ Solved examples

1. In the Fibonacci sequence 1, 1, 2, 3, 5, 8, ..., find the 10th term.
Continue: 1,1,2,3,5,8,13,21,34,55. The 10th term is 55.
2. Find the sum of the first 8 Fibonacci numbers (1,1,2,3,5,8,13,21).
Use F1+...+Fn = F(n+2) − 1. Here F10 = 55, so sum = 55 − 1 = 54. (Check: 1+1+2+3+5+8+13+21 = 54.)
3. A sequence is defined by a1 = 2 and a(n) = 3·a(n−1) + 1. Find a4.
a1 = 2, a2 = 3·2+1 = 7, a3 = 3·7+1 = 22, a4 = 3·22+1 = 67.
4. Find the sum S = 1 + 2·(1/2) + 3·(1/2)² + 4·(1/2)³ + ... (infinite AGP).
AGP with a = 1, d = 1, r = 1/2. S∞ = a/(1−r) + dr/(1−r)² = 1/(1/2) + (1·½)/(½)² = 2 + (1/2)/(1/4) = 2 + 2 = 4.

✏️ Practice — try these, take hints as needed

1. In Fibonacci (1,1,2,3,5,...), find the 8th term.
Each term = sum of previous two.
Continue: 1,1,2,3,5,8,13,...
List up to the 8th.
21
2. Find the sum of the first 10 Fibonacci numbers.
Use F1+...+Fn = F(n+2) − 1.
You need F12.
Fibonacci: ...,55,89,144 ⇒ F12 = 144.
143
3. a1 = 1, a(n) = 2·a(n−1) + 3. Find a4.
Compute term by term.
a2 = 2·1+3 = 5.
a3 = 2·5+3 = 13, then a4.
29
4. Find S = 1 + 2·(1/3) + 3·(1/3)² + 4·(1/3)³ + ... (infinite).
AGP with a=1, d=1, r=1/3.
S∞ = a/(1−r) + dr/(1−r)².
1/(2/3) + (1/3)/(2/3)².
9/4
5. A sequence repeats with a1 = 3, a2 = 5, and a(n) = a(n−1) − a(n−2). Find a7.
Compute terms: 3,5,2,−3,−5,−2,...
Note the pattern repeats every 6 terms.
a7 equals a1.
3

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Special Series Take the chapter test →