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Joe Harper Joe Harper Dec 21, 2020

17. The sum of the first and 8 terms and first 24 terms of an AP are equal. Findthe sum of its 32 terms.​

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Answer:

{\huge{\underline{\underline{\rm{\red{Question:-}}}}}}

Q- The sum of first 8 terms and first 24 terms of an A.P. are equal . Find the sum of first 32 terms .

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{\huge{\underline{\underline{\pink{\rm{Answer\checkmark}}}}}}

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♡ Concept :

• In these type of Questions , we Simply derive relation between first term(a) and common Difference (d) .

• In this question , we will compare 8th and 24th term .

• Then we will get a relation between a and d .

• Then we will use that relation , to find the sum of first 32 terms .

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♡ Given :
  •  \rm \: s _{8} = s _{24}
  • It's an A.P.
♡ To Find :
  •  \rm \: s _{32}
♡ Solution :
  • a = first term .
  • d = common difference
  •  \rm \: s _{n}= sum \: of \: n \: terms

Since , we know that the sum of first 8 terms is equal to the sum of first 24 terms

so ,

equating them ..

 \rm \: s_{8} = s _{24}

 \rm \mapsto \:\frac{ \cancel8}{ \cancel2} (2a + (8 - 1)d) =\frac{ \cancel{24}} {\cancel{2}} (2a + (24 - 1)d) \\

 \mapsto \rm \:\cancel4(2a + 7d) = \cancel{ 12}(2a + 23d)

 \mapsto \rm \: 2a + 7d = 3(2a + 23d)

 \mapsto \rm \: 2a + 7d = 6a + 69d

 \mapsto \rm \: 6a - 2a + 69d - 7d = 0

 \mapsto { \rm \: 4a+62d = 0 }

 \mapsto \boxed{ \rm \: 2a + 31d = 0}.....(i)

Now ,

We have to find the sum of first 32 terms ..

so , using the Formula

 \rm \: s_{32} =\frac{32}{2} (2a + (32 - 1)d) \\

 \mapsto \rm \: s _{32} = 16(2a + 31d)

From eq(i) ,

  • 2a + 31d = 0

 \mapsto \rm \: s _{32} = 16 \times 0

 \large \boxed{ \rm \: s _{32} = 0}

So , the sum of first 32 terms would be 0 .

_________________________♡ Know More :

1. \rm \: sum \: of \: n \: terms \:=\frac{n}{2} (2a + (n - 1)d) \\

2. \rm \: n \: th \: term \:= a + (n - 1)d

Where ,

 \pink{ \ddot{ \smile}} \rm \: a = first \: term \:

 \rm \blue{ \ddot{ \smile}} \: d = common \: difference \:

 \green { \ddot{ \smile}} \rm \: n ={n}^{th}\: term \:

174
Nadezhda Nadezhda
Dec 21, 2020
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