Question

the sum of the 3 rd and 7 th terms of an A.P is 6 and their product is 8;. find the 1 st term and the common difference of an Ap​

Answer 1

Answer: First term of A.P is 1 and common difference is 1/2

or First term of A.P is 5 and common difference is -1/2

Step-by-step explanation:

Let f be the first term and d be the common difference

an=f+(n-1)d

Therefore,a3=f+(3-1)d=f+2d

a7=f+(7-1)d=f+6d

 a3+a7=(f+2d)+(f+6d)=2f+8d

 2f+8d=6

By simplifying, f+4d=3       -(equation 1)

a3*a7=(f+2d)(f+6d)

        =(f+4d-2d)(f+4d+2d)

        =(3-2d)(3+2d)          from equation 1

hence,(3-2d)(3+2d)=8

9-4$d^{2}$=8

4$d^{2}$=1

$d^{2}$=1/4

d=1/2 or -1/2

If d=1/2

f+4*1/2=3    from equation1

f=1

if d=-1/2

f+4*(-1/2)=3    from equation1

f=5

Therefore, First term of A.P is 1 and common difference is 1/2

or First term of A.P is 5 and common difference is -1/2

Answer 2

Answer:

3rd term is a+2d and 7th term is a+6d

Now sum a+2d+a+6d=6

2a+8d= 6

a+4d= 3 equation (1)

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

a²+6ad+2ad+12d²= 8

a²+8ad+12d²= 8

Now put. a=3-4d. from equation 1

so. (3-4d)²+8(3-4d)d+12d²= 8

9+16d²-24d+24d-32d²+12d²= 8

9-4d²= 8

9-8= 4d²

1= 4d²

1/4= d²

d= 1/2

Now put d=1/2 in equation 1 (a+4d=3)

a+4x1/2= 3

a+2= 3

a= 1

Hence,

First term (a) = 1

Common difference (d)= 1/2