Question

The 12th term of a parallel series is -13 and the sum of its first 4 terms is 24. Find the sum of the first 10 terms in this series.

Answer 1

Given :-

12th term of the series is - 13

Sum of its first 4 terms = 24

Required to find :-

• Sum of first 10 terms of the series ?

Formula used :-

$\huge{\dagger{\boxed{\rm{ {S}_{nth} = \dfrac{n}{2} [ 2a + ( n - 1 ) d ] }}}}$

SoLuTioN :-

Given that :-

12th terms of the series = - 13

Sum of its first 4 terms = 24

we need to find the sum of first 10 terms of that sequence .

So,

12th term = - 13

But 12th term can be represented as " a + 11d "

So,

a + 11d = - 13 $\rightarrowtail{\text{Equation-1}}$

Consider this as equation - 1

Similarly,

It is also mentioned that ;

Sum of first 4 terms = 24

But this is actually written as ;

$\tt{ {S}_{4} = \dfrac{4}{2} [ 2a + ( 4 - 1 )d ]}$

$\tt{ \bigg( \; where , \; {S}_{nth} = {S}_{4} \; \bigg)}$

But ,

$\tt{ \bigg[ {S}_{4} = 24 \bigg] }$

So,

$\rm{ 24 = 2 [ 2a + 3d ] }$

$\rm{ 24 = 4a + 6d }$

$\rm{ 4a + 6d = 24 }$$\longrightarrow{\text{Equation -2 }}$

Consider this as equation - 2

Now ,

Multiply equation 1 with 4

4 ( a + 11d ) = 4 ( - 13 )

4a + 44d = - 52 $\rightarrowtail{\text{Equation - 3 }}$

Consider this as equation 3

Hence,

Subtract equation 2 from equation 3

4a + 44d = -52

4a + 6d = 24

( - ) ( - ) ( - )

$\rule{75}{1}$

0 + 38d = -76

$\rule{75}{1}$

This implies;

38d = - 76

$\tt{ d = \dfrac{-76}{38}}$

d = - 2

Substitute the value of d in Equation 1

=> a + 11d = - 13

=> a + 11 ( - 2 ) = - 13

=> a - 22 = - 13

=> a = - 13 + 22

a = 9

Now let's find the sum of first 10 terms of the series

Using the formula;

$\huge{\dagger{\boxed{\rm{ {S}_{nth} = \dfrac{n}{2} [ 2a + ( n - 1 ) d ] }}}}$

So,

$\rightarrowtail{\rm{ {S}_{nth} = {S}_{10} }}$

$\rightarrowtail{\rm{ {S}_{10} = \dfrac{10}{2} [ 2 ( 9 ) + ( 10 - 1 ) -2 ]}}$

$\rightarrowtail{\sf{ {S}_{10} = 5 [ 18 + ( 9 ) -2 ]}}$

$\rightarrowtail{\sf{ {S}_{10} = 5 [ 18 - 18 ]}}$

$\rightarrowtail{\sf{ {S}_{10} = 5 \times 0 }}$

$\rightarrowtail{\sf{ {S}_{10} = 0 }}$

Therefore,

Sum of first 10 terms of the series = 0