Question

In an AP,sum of first 8 terms is 136 and that of 15 terms is 465. Find the sum of 25 terms.​

Answer 1

Solution :

$\bigstar$ Using formula of the sum of an A.P;

$\boxed{\bf{S_n=\frac{n}{2} \bigg[2a+(n-1)d\bigg]}}}}$

• a is the first term
• d is the common difference
• n is the term of an A.P.

A/q

$\longrightarrow\sf{136=\cancel{\dfrac{8}{2}} \bigg[2a+(8-1)d\bigg]}\\\\\longrightarrow\sf{136=4[2a+7d]}\\\\\longrightarrow\sf{\cancel{136/4}=2a+7d}\\\\\longrightarrow\sf{34=2a+7d........................(1)}$

&

$\longrightarrow\sf{465=\cancel{\dfrac{15}{2}} \bigg[2a+(15-1)d\bigg]}\\\\\longrightarrow\sf{930=15[2a+14d]}\\\\\longrightarrow\sf{\cancel{930/15}=2a+14d}\\\\\longrightarrow\sf{62=2a+14d}\\\\\longrightarrow\sf{62=2(a+7d)}\\\\\longrightarrow\sf{\cancel{62/2} = a+7d}\\\\\longrightarrow\sf{31=a+7d.......................(2)}$

$\underline{\boldsymbol{Using\:by\:Substitution\:method\::}}}$

From equation (2),we get;

$\longrightarrow\sf{a=31-7d.....................(3)}$

∴Putting the value of a in equation (1),we get;

$\longrightarrow\sf{34=2(31-7d) +7d}\\\\\longrightarrow\sf{34=62 - 14d + 7d}\\\\\longrightarrow\sf{34=62-7d}\\\\\longrightarrow\sf{34-62=-7d}\\\\\longrightarrow\sf{-28=-7d}\\\\\longrightarrow\sf{d=\cancel{-28/-7}}\\\\\longrightarrow\bf{d=4}$

∴Putting the value of d in equation (3),we get;

$\longrightarrow\sf{a=31-7(4)}\\\\\longrightarrow\sf{a=31-28}\\\\\longrightarrow\bf{a=3}$

Now;

$\longrightarrow\sf{S_{25}=\dfrac{25}{2} \bigg[2(3) + (25-1)(4) \bigg]}\\\\\\\longrightarrow\sf{S_{25} = \dfrac{25}{2} \bigg[6 + 24\times 4 \bigg]}\\\\\\\longrightarrow\sf{S_{25}=\dfrac{25}{2} \bigg[6+96\bigg]}\\\\\\\longrightarrow\sf{S_{25} = \dfrac{25}{\cancel{2}} \times \cancel{102}}\\\\\longrightarrow\sf{S_{25} = 25\times 51}\\\\\longrightarrow\bf{S_{25} = 1275}$

Thus;

The sum of 25 terms will be 1275 .

Answer 2

*۰۪۫G۪۫۰۰۪۫i۪۫۰۰۪۫v۪۫۰۰۪۫e۪۫۰۰۪۫n۪۫۰:

• Sum of first 8 terms(S8)=136

• Sum of 15 terms (S15) =465

* ۰۪۫T۪۫۰۰۪۫o۪۫۰ ۰۪۫F۪۫۰۰۪۫i۪۫۰۰۪۫n۪۫۰۰۪۫d۪۫۰:

• Sum of 25 terms(S25) =?

* ۰۪۫S۪۫۰۰۪۫o۪۫۰۰۪۫l۪۫۰۰۪۫u۪۫۰۰۪۫t۪۫۰۰۪۫i۪۫۰۰۪۫o۪۫۰۰۪۫n۪۫۰

Formula for sum of an A. P :

_____________

Sn= n/2[2a+(n-1) d]

_____________

Now to find the sum of first 8 terms,

By Using formula,

S8=8/2[2a+(8-1) d]

→136=4[2a+7d]

→136/4=2a+7d

→34=2a+7d

→2a+7d=34...................... (1)

Now to find the Sum of 15 terms,

By using formula,

S15=15/2[2a+(15-1) d]

→465=15/2[2a+14d]

→465×2/15=2a+14d

→ 31×2=2a+14d

→ 62=2a+14d

→62=2(a+7d)

→62/2=a+7d

→31=a+7d

→a+7d=31........................(2)

Subtracting equation (2) From (1) we get,

a=3

Now Substituting the value of a in (2)

→3+7d=31

→7d=31-3

→7d=28

→d=28/7

→d=4

Now, Let's find the Sum of 25 terms.

By using formula,

S25=25/2[2×3+(25-1) 4

=25/2[6+24×4]

=25/2[6+96]

=25/2×102

=25×51

=1275

Therefore, The sum of 25 terms is 1275.