A series whose terms are in Geometric progression is called Geometric series.

**Sum
to n terms of a G.P.**

A series whose terms are
in Geometric progression is called Geometric series.

Let *a*, *ar*,
*ar*^{2} , ...*ar ^{n}*

The sum of first *n*
terms of the Geometric progression is

*S _{n} *=

Multiplying both sides
by *r*, we get *rS _{n}* =

(2)−(1) gives *rS _{n}* −

*S _{n}* (

Thus, the sum to n terms
is* *

**Note**

The above formula for
sum of** **first *n* terms of a G.P. is not applicable when *r* =
1.

If *r* = 1 , then

*S _{n} *=

**Progress Check**

1. A series whose
terms are in Geometric progression is called _______.

2. When*r* = 1 ,
the formula for finding sum to *n* terms of a G.P. is ______.

3. When*r* ≠ 1 ,
the formula for finding sum to n terms of a G.P. is ______.

The sum of infinite
terms of a G.P. is given by *a* + *ar* +*ar* ^{2}
+ *ar* ^{3} +….. = *a* /(1-r), -1 < r < 1

**Example 2.46 **Find the sum of** **8** **terms of the G.P.** **1,** **−** **3, 9,** **−27…

*Solutions*

Here the first term *a*
= 1 , common ratio *r* = -3/1 = -3 < 1, Here *n* = 8.

Sum to *n* terms of
a G.P. is

Find the first term of a
G.P. in which** ***S*_{6 }= 4095 and *r* = 4.

Common ratio = 4 > 1
, Sum of first 6 terms *S*_{6} = 4095

First term *a *=*
*3* *.

How many terms of the
series**
**1** **+** **4** **+** **16** **+** **make the sum** **1365 ?

Let* **n*** **be the number of terms
to be added to get the sum

4* ^{n}*
= 4096 then 4

*n* = 6

Find the sum

** Solution **Here

Sum of infinite terms =

Find the rational form
of the number** **0.6666¼

We can express the
number**
**0.6666¼as follows

0.6666… = 0.6 + 0.06 +
0.006 + 0.0006 +

We now see that numbers
0.6, 0.06, 0.006 ... forms an G.P. whose first term *a* = 0.6

and common ration *r*
= 0.06 / 0.6 = 0.1 . Also − 1 < *r* = 0.1 < 1

Using the infinite G.P.
formula, we have

0.6666... = 0.6 + 0.06 + 0.006 + 0.0006 + ... =

Thus the rational number
equivalent of 0.6666 is 2/3

Find the sum to** ***n*** **terms of the series** **5** **+** **55** **+** **555** **+ ...

The series is neither
Arithmetic nor Geometric series. So it can be split into two** **series and then find the
sum.

Find the least positive
integer**
***n*** **such that** **1** **+** **6** **+** **6^{2}** **+ +** **6^{n}** **>** **5000

We have to find the
least number of terms for which the sum must be greater** **than 5000.

That is, to find the
least value of *n*. such that *S _{n} *>

We have

6* ^{n}*
-1 > 25000 gives 6

Since,
6^{5} = 7776 and 6^{6 }=46656

The least positive value
of *n* is 6 such that 1 + 6 + 6^{2} + + 6* ^{n}*
> 5000

**Example 2.53**

A person saved money
every year, half as much as he could in the previous** **year. If he had totally
saved ₹ 7875 in 6 years then how much did he save in the first year?

Total amount saved in** **6

Since he saved half as
much money as every year he saved in the previous year,

We have *r* = ½
< 1

The amount saved in the
first year is ₹ 4000.

Tags : Theorem, Example, Solution | Mathematics , 10th Mathematics : Numbers and Sequences

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10th Mathematics : Numbers and Sequences : Sum to n/infinite terms of a G.P.(Geometric Progression) | Theorem, Example, Solution | Mathematics

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