# Arithmetic and Geometric and Harmonic Sequences Calculator

Determine SequenceExpand Sequence
<-- Enter Series
<-- (Optional) Number of Expansion terms
<-- Enter First term a1
<-- Enter d

Calculate the explicit formula, term number 10, and the sum of the first 10 terms for the following geometric series:
2,4,8,16,32

The explicit formula for a geometric series is an = a1r(n - 1)
r represents the common ratio between each term below:
 r = an an - 1

Looking at all the terms, we see the common ratio (r) is 2, and we have a1 = 2
We simplify a1 to just a
ar = 2 x 2 =
Therefore, our explicit formula is an = (2)2(n - 1)

Calculate Terms (6 - 10)
Using our explicit formula with n = 10 and r = 2, we have:
Term #Math Step 1Math Step 2Math Step 3Term
6a6 = 2 x 2(6 - 1)a6 = 2 x 25a6 = 2 x 32a6 = 64
7a7 = 2 x 2(7 - 1)a7 = 2 x 26a7 = 2 x 64a7 = 128
8a8 = 2 x 2(8 - 1)a8 = 2 x 27a8 = 2 x 128a8 = 256
9a9 = 2 x 2(9 - 1)a9 = 2 x 28a9 = 2 x 256a9 = 512
10a10 = 2 x 2(10 - 1)a10 = 2 x 29a10 = 2 x 512a10 = 1024

Calculate the sum of the first 10 terms of the sequence, denoted Sn:
 Sn = a1(1 - rn) 1 - r

With n = 10, and r = 2, we get:
 S10 = 2(1 - 210) 1 - 2

 S10 = 2(1 - 1024) -1

 S10 = 2(-1023) -1

 S10 = -2046 -1

S10 = 2046