Question

The sum of 8 terms is 136 and that of 15 terms is 465 .Then find the sum of first 25 terms

Answer 1

Answer:

S25 =1275

to find the sum of the first 25 terms ;

if the sum of the first 8terms of AP is 136 and that of first 15 terms is 465.

we know that sum of first n terms of an AP having the first term 'a' and common difference'

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

Ss =8/2 (2a+(8-1)d)

Ss=4(2a+7d)

136/4 =2a +7d

2a+7d =34....eq1

by the same way

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

S15=15/2(2a+14d)

465*2/15=2a+14d

2a+14d=62....eq2

now subtract both equations and find the value of a and d

2a+7d=34

2a+14d=62

-7d=-28

d=4

thus

2a+7d=34

2a+28=34

2a=34-28

a=3

so, sum of first 25 terms

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

S25=25/2(6+24*4)

S25=25/2 *102

S25=1275

hope it helps you

Answer 2

Step-by-step explanation:

AnswEr:

$\small\bold{\underline{\sf{\blue{\:\:Given\;\::-}}}}$

Sum of the 8th Terms is 136 and that of first 15 terms is 465.

To Find:-

Sum of first 25 Terms = ?

Explanation:-

We know that,

$\bigstar\;\:\small{\underline{\boxed{\sf{\red{\dfrac{n}{2} 2a \:+\:(n\:-\:1)d}}}}}$

Here,

a = First Term

D = Common Difference

$\bold{\underline{\sf{According\:to\:Ques\: Now}}}$

$\longrightarrow\sf\: \dfrac{8}{2} (2a \:+\:(8\:-\:1)d \:=\: 136$

$\longrightarrow\sf\: 2a + 7d = 136$ -------Eq(1)

$\rule{150}2$

$\longrightarrow\sf\: \dfrac{15}{2} (2a \:+\:(15\:-\:1)d = 465$

$\longrightarrow\sf\: 2a \:+\:14d \:=\:\dfrac{465\times\: 2}{15}$

$\longrightarrow\sf\: 2a + 14d = 62$ -------Eq(2)

$\rule{150}2$

$\dag\:\small\bold{\underline{\sf{\blue{Now,\: Subtracting\: Equation\:(1)\:from\:(2)}}}}$

$\longrightarrow\sf\: 2a \: + \: 17d \: =\: 136$

$\longrightarrow\sf\:2a \:+\: 14d \:=\: 62$

$\longrightarrow\sf\: - 7d = - 28$

$\longrightarrow\sf\: d = \dfrac{-28}{-7}$

$\longrightarrow\large\boxed{\sf{\red{d\:=\: 4}}}$

★ Substituting the Value of d in Equation (1)

$\longrightarrow\sf\: 2a + 7d = 34$

$\longrightarrow\sf\:2a + 7(4) = 34$

$\longrightarrow\sf\:2a + 28 = 34$

$\longrightarrow\sf\:2a = 34 - 28$

$\longrightarrow\sf\:2a = 6$

$\longrightarrow\sf\:a = \cancel\dfrac{6}{2}$

$\longrightarrow\large{\sf{\red{a \:=\: 3}}}$

$\rule{150}2$

Now, Finding the Sum of 25 Terms

$\longrightarrow\sf\: Sn = \dfrac{n}{2}(2a \:+\;(n \:-\:1)d$

$\longrightarrow\sf\: = \dfrac{25}{2}(6 + 24 \times \: 4)$

$\longrightarrow\sf\: \dfrac{25}{2} \times\: 102$

$\longrightarrow\large{\underline{\boxed{\sf{\pink{1275}}}}}$

$\small\bold{\underline{\sf{\blue{Hence,\;Sum\:of\: First\:25\;terms\: is\:1275}}}}$