The sum of third and seventh of an A.P is 6 and their product is 8. Find the first term and the common difference of the A.P.
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Answers
Answer:if common difference is 0.5 then first term =1
AND
if common difference is (-0.5) then first term=5
Step-by-step explanation:
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 ......(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 .....(2)
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
there fore common difference is 0.5 OR -0.5
and from 1 and 2
we get a=1 or a=5