Sequence Calculator
Find the nth term of arithmetic, geometric, and Fibonacci sequences. Enter the first term, common difference or ratio, and term number.
What is the difference between a sequence and a series?
A sequence is an ordered list of numbers (e.g., 2, 4, 6, 8). A series is the sum of a sequence (e.g., 2+4+6+8=20). Sequence shows individual terms, series shows their total. Example: Fibonacci sequence: 1,1,2,3,5,8,13... Sum of first 5 Fibonacci numbers = 1+1+2+3+5 = 12 (series).
How do you find the nth term of an arithmetic sequence?
Formula: a_n = a_1 + (n-1)d, where a_1 is first term, d is common difference, n is term number. Example: For sequence 5, 8, 11, 14... (d=3), find 15th term: a_15 = 5 + (15-1)*3 = 5 + 42 = 47. Quick check: each term adds 3, so 14 terms after 5 gives 47.
How do you find the nth term of a geometric sequence?
Formula: a_n = a_1 * r^(n-1), where a_1 is first term, r is common ratio, n is term number. Example: For sequence 3, 6, 12, 24... (r=2), find 8th term: a_8 = 3 * 2^(8-1) = 3 * 128 = 384. Each term doubles, so 3→6→12→24→48→96→192→384.
What is the Fibonacci sequence and how is it calculated?
Fibonacci sequence: each term is sum of previous two terms. Starts 0,1,1,2,3,5,8,13,21,34... Formula: F_n = F_(n-1) + F_(n-2), with F_0=0, F_1=1. Example: F_6 = F_5 + F_4 = 5 + 3 = 8. Found in nature (spirals, petals), architecture, art. Ratio of consecutive terms approaches golden ratio rho� ≈ 1.618.
How do you identify the pattern in a sequence?
Check differences between consecutive terms (arithmetic: constant difference), ratios (geometric: constant ratio), or other patterns. Example: 2,5,8,11 → differences 3,3,3 → arithmetic. 2,6,18,54 → ratios 3,3,3 → geometric. 1,1,2,3,5,8 → each = sum of previous two → Fibonacci.