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If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms. Q.9

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The best way for solving the problem of arithmetic progressions of ncert of exercise 5.3. How i solve this problem please guide me the best way If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.

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1. Given that,

S7Â = 49

S17Â = 289

We know, Sum of n terms;

SnÂ =Â n/2Â [2aÂ + (nÂ â€“ 1)d]

Therefore,

S7=Â 7/2Â [2aÂ +(nÂ -1)d]

S7Â = 7/2Â [2aÂ + (7 -1)d]

49 = 7/2Â [2aÂ + 6d]

7 = (a+3d)

aÂ + 3dÂ = 7 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.Â (i)

In the same way,

S17Â = 17/2Â [2a+(17-1)d]

289 = 17/2 (2aÂ +16d)

17 = (a+8d)

aÂ +8dÂ = 17 â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.Â (ii)

Subtracting equationÂ (i)Â from equationÂ (ii),

5dÂ = 10

dÂ = 2

From equationÂ (i), we can write it as;

a+3(2) = 7

a+Â 6 = 7

a =Â 1

Hence,

SnÂ =Â n/2[2a+(n-1)d]

=Â n/2[2(1)+(nÂ â€“ 1)Ã—2]

=Â n/2(2+2n-2)

=Â n/2(2n)

=Â n2

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