# 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.

2
I want answer as it is urgent

2014-10-31T06:45:38+05:30
Let the first term be a and common difference be d
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
Nice!!
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well yea ur rite.
2014-10-31T12:59:00+05:30
So using formula for n th term

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