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 n^{th} 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

**a**_{n} = a_{1}*r^{n-1}

To calculate the sum of a geometric sequence

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

Where r is the common ratio

n is the number of terms to find

a_{1} is the 1st term of the sequence

S_{n} is the sum of the n-terms.

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

To calculate the sum of a geometric sequence

S

Where r is the common ratio

n is the number of terms to find

a

S

Geometric Sequence Formula's:

To find 'nth' term of a geometric sequence:

a_{n} = a_{1} * r^{n-1}

To find sum of geometric sequence:

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

Where

a = first term of the sequence

r = common ratio

n = number of term

a_{n} = n^{th} term of a geometric sequence

s_{n} = sum of geometric sequence

Put the values in the formulas and calculate it further.

Given the terms a

_{3}= 216 and a_{6}= 46656 of a geometric sequence, find the exact value of the term a_{9}of the geometric sequence and sum of the geometric sequence.**Step 1 :**Given: The Terms a

_{3}= 216 and a_{6}= 46656 of a geometric sequence,

To find 'nth' term of a geometric sequence:

a

_{n}= a_{1}* r^{n-1}

To find sum of geometric sequence:

s

_{n}= $\frac{(a_{1}(1-r^{n}))}{(1 - r)}$**Step 2 :**Put the values in the formula and calculate it further.

a

_{3}= a_{1}* r^{3-1}= 216

a_{6}= a_{1}* r^{7-1}= 46656

Divide the terms a_{3}and a_{6}to write

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

Solve for r to obtain.

r^{3}= 216 which gives**r =6**

Use a_{3}to find a_{1}as follows.

a_{3}= 216 = a_{1}(6)^{2}

Solve for a_{1}to obtain.

**a**_{1}= 6

**n = 9**

a

_{9}= 6(6)^{8}= 10077696

Use the sum formula

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

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

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

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

= 12093234

**Answer :****Term of poistion a**= 10077696_{9}**Sum of geometric sequence**= 12093234Given the terms a

_{3}= 125 and a_{6}= 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 a

_{3}= 125 and a_{6}= 15625 of a geometric sequence,

To find 'nth' term of a geometric sequence:

a

_{n}= a_{1}* r^{n-1}

To find sum of geometric sequence:

s

_{n}= $\frac{(a_{1}(1-r^{n}))}{(1 - r)}$**Step 2 :**Put the values in the formula and calculate it further.

a

_{3}= a_{1}* r^{3-1}= 125

a_{6}= a_{1}* r^{6-1}= 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.

r^{3}= 125 which gives**r = 5**

Use a

_{3}to find a_{1}as follows.

a_{3}= 125 = a_{1}(5)^{2}

Solve for a_{1}to obtain.

**a**_{1}= 5

**n = 10**

a_{10}= 5(5)^{9}= 9765625

Use the sum formula

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

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

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

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

= 12207030

**Answer :****Term of poistion a**= 9765625_{9}**Sum of geometric sequence**= 12207030

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