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Geometric Sequence Calculator
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Sequence is the list of ordered terms and the sum of these ordered terms is known as Series. Geometric Sequence Calculator or Geometric Series Calculator is an online tool used to find the nth term and sum of geometric sequence.

Geometric sequence is a sequence of numbers such that the ratio of any two successive members of the sequence is a constant. Sequence is also called as progression.
To calculate any term of a geometric sequence
an = a1*rn-1

To calculate the sum of a geometric sequence
Sn = $\frac{a_{1}(1 - r^{n})}{(1 - r)}$

Where r is the common ratio
n is the number of terms to find
a1 is the 1st term of the sequence
Sn is the sum of the n-terms.

## Step by Step Calculation

Step 1 :

Geometric Sequence Formula's:

To find 'nth' term of a geometric sequence:

an = a1 * rn-1

To find sum of geometric sequence:

sn = $\frac{(a_{1}(1-r^{n}))}{(1 - r)}$

Where

a = first term of the sequence

r = common ratio

n = number of term

an = nth term of a geometric sequence

sn = sum of geometric sequence

Step 2 :

Put the values in the formulas and calculate it further.

## Example Problems

1. ### Given the terms a3 = 216 and a6 = 46656 of a geometric sequence, find the exact value of the term a9 of the geometric sequence and sum of the geometric sequence.

Step 1 :

Given: The Terms a3 = 216 and a6 = 46656 of a geometric sequence,

To find 'nth' term of a geometric sequence:

an = a1 * rn-1

To find sum of geometric sequence:

sn = $\frac{(a_{1}(1-r^{n}))}{(1 - r)}$

Step 2 :

Put the values in the formula and calculate it further.

a3 = a1 * r3-1 = 216

a6 = a1 * r7-1 = 46656

Divide the terms a3 and a6 to write

$\frac{(a_{6})}{(a_{3})}$ = $\frac{(a_{1}* r^{5})}{(a_{1}*r^{2})}$ = $\frac{46656}{216}$

Solve for r to obtain.

r3 = 216 which gives r =6

Use a3 to find a1 as follows.

a3 = 216 = a1(6)2

Solve for a1 to obtain.

a1 = 6

n = 9

a9 = 6(6)8 = 10077696

Use the sum formula

sn = $\frac{(a_{1}(1-r^{n}))}{(1 - r)}$

= $\frac{(6(1-6^{9}))}{(1-6)}$

= $\frac{(6(1-10077696))}{(-5)}$

= $\frac{(6(10077695))}{5}$

= 12093234

Term of poistion a9 = 10077696

Sum of geometric sequence = 12093234

2. ### Given the terms a3 = 125 and a6 = 15625 of a geometric sequence, find the exact value of the term a10 of the sequence and sum of the sequence.

Step 1 :

Given: he terms a3 = 125 and a6 = 15625 of a geometric sequence,

To find 'nth' term of a geometric sequence:

an = a1 * rn-1

To find sum of geometric sequence:

sn = $\frac{(a_{1}(1-r^{n}))}{(1 - r)}$

Step 2 :

Put the values in the formula and calculate it further.

a3 = a1 * r3-1 = 125

a6 = a1 * r6-1 = 15625

Divide the terms a3 and a6 to write

$\frac{(a_{6})}{(a_{3})}$ = $\frac{(a_{1}*r^{5})}{(a_{1}*r^{2})}$ = $\frac{156251}{25}$

Solve for r to obtain.

r3 = 125 which gives r = 5

Use a3 to find a1 as follows.

a3 = 125 = a1(5)2

Solve for a1 to obtain.

a1 = 5

n = 10

a10 = 5(5)9 = 9765625

Use the sum formula

sn = $\frac{(a_{1}(1-r^{n}))}{(1-r)}$

= $\frac{(5(1-5^{10}))}{(1-5)}$

= $\frac{(5(1-9765625))}{(1-5)}$

= $\frac{(5(9765624))}{4}$

= 12207030

Term of poistion a9 = 9765625

Sum of geometric sequence = 12207030

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