# The sum of the third and the seventh terms of an Ap is 6 and their product is 8.find the sum of first sixteen term of ap.

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by Yuktadimpu

plz give me the answer

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by Yuktadimpu

plz give me the answer

I want answer as it is urgent

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The Brainliest Answer!

nth term = a+(n-1)d

Given Third Term + Seventh term = (a+2d)+(a+6d) = 6 ==> a+4d = 3

hence, a= 3-4d

Third Term * Seventh term = (a+2d)*(a+6d) = 8

(3-4d+2d)*(3-4d+6d) = 8==> (3-2d)*(3+2d) = 8

i.e. 9-4d^2 = 8==> d^2 = (9-8)/4 = 0.25==> d = 0.5 or -0.5

Now to check which is correct d...

Substitute and find

Case (a): d= 0.5

a+4d = 3==> a=3-4d = 3-4(0.5)=1

3rd term = a+2d= 1+2*0.5 = 2

7th term = a+6d= 1+6*0.5 = 4

Sum = 6 and Product = 8

Case (b): d= -0.5

a+4d = 3==> a=3-4d = 3-4(-0.5) = 3+2 = 5

3rd term = a+2d= 5+2*(-0.5) = 4

7th term = a+6d= 5+6*(-0.5) = 2

Sum = 6 and Product = 8

Since both are matching, we will go with bothvalues

Sum of first 16 terms = n*(2a+(n-1)d)/2 = 16*(2a+15d)/2

= 8*(2a+15d)

Case (a): d= 0.5

Sum = 8*(2*1+15*0.5)=76

Case (b): d= 0.5

Sum = 8*(2*5+15*(-0.5))=20

a = first term

n = no. of terms

d = common difference

So given

a + (3-1)d + a (7-1)d = 6

2a + 8d = 6

a + 4d = 3

a = 3 - 4d

(a + 2d)(a + 6d) = 8

(3 - 2d)( 3 + 2d) = 8 {substituting a = 3 - 2d}

9 - 4d² = 8

d = +1/2 and - 1/2

so when d = +1/2 then a = 3 - 2 = 1 and when d = -1/2 a = 3 + 2 = 5

so using formula for

when d = +1/2

Similarly when d = -1/2