 Questions & Answers    Question Answers

# The first term of an infinite G.P. is $1$ and any term is equal to the sum of all the succeeding terms. Find the sum of the infinite series.  Answer Verified
Hint: Any term in G.P. is equal to the sum of all the succeeding terms. We have:
$\Rightarrow {T_n} = {T_{n + 1}} + {T_{n + 2}} + {T_{n + 3}} + ......\infty$

The general term of G.P. can be written as:
$\Rightarrow {T_n} = a{r^{n - 1}} .....(i)$
And according to the information given in the question, any term of the G.P. is equal to the sum of all the succeeding terms. From this we’ll get:
$\Rightarrow {T_n} = {T_{n + 1}} + {T_{n + 2}} + {T_{n + 3}} + ......\infty$
Substituting corresponding values in equation$(i)$, we’ll get:
$\Rightarrow a{r^{n - 1}} = a{r^n} + a{r^{n + 1}} + a{r^{n + 2}} + ......\infty ,$
$a$ is the first term of G.P. and its value is $1$ as per the information given in the question. So putting its value, we’ll get:
$\Rightarrow {r^{n - 1}} = {r^n} + {r^{n + 1}} + {r^{n + 2}} + .....\infty , \\ \Rightarrow {r^{n - 1}} = {r^n}\left[ {1 + r + {r^2} + .....\infty } \right], \\ \Rightarrow \dfrac{1}{r} = \left[ {1 + r + {r^2} + .....\infty } \right] .....(ii) \\$
Now, the terms on the right hand side of the above equation constitutes an infinite G.P. with $1$ as the first term and $r$ as the common ratio. And we know the formula for sum of terms of infinite G.P.:
$\Rightarrow {S_\infty } = \dfrac{a}{{1 - r}}$
So, on using this formula for equation $(ii)$,we’ll get:
$\Rightarrow \dfrac{1}{r} = \dfrac{1}{{1 - r}}, \\ \Rightarrow 1 - r = r, \\ \Rightarrow 2r = 1, \\ \Rightarrow r = \dfrac{1}{2}. \\$
Thus, the common ratio of the G.P. is $\dfrac{1}{2}$ and its first term is already given as $1$. So, we our infinite G.P.:
$\Rightarrow 1,\dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{8},.......\infty$
For finding sum of its terms, we will again apply${S_\infty } = \dfrac{a}{{1 - r}}$, we’ll get:
$\Rightarrow {S_\infty } = \dfrac{1}{{1 - \dfrac{1}{2}}}, \\ \Rightarrow {S_\infty } = 2 \\$
Therefore, the sum of infinite G.P. is $2$.

Note: If a G.P. consists of infinite terms, then we can only calculate the sum of its terms if it's common ratio is greater than $0$ and less than $1$$\left( {0 < r < 1} \right)$.Otherwise its sum will not be defined.
Bookmark added to your notes.
View Notes
Sequences and Series  Finite and Infinite Sets  Expression Term Factor Coefficient  Infinite Solutions  Geometric Progression Sum of GP  Maxima and Minima - Using First Derivative Test  What is an Alloy? - Definition and Examples  Sequence and Series  Cleaning Capacity of any Soap in Hard and Soft Water  The Difference Between an Animal that is A Regulator and One that is A Conformer  