Question

The sum of third and seventh term of an A. P. is 6 and their product is 8. Find the first term and the common difference of the A. P. 

Answer 1

`t_n=a(n-1)d` 

∴ `t_3=a+(3-1)d=a+2d` 

`t_7=a+(3-1)d=a+2d`

∴ `t_3+t_7=(a+2d)+(a+6d)=2a+8d` 

∴ `2a+8d=6` 

∴ a+4d=3 ................(I) 

`t_3xxt_7=(a+2d)(a+6d)` 

            = `(a+4d-2d) (a+4d+2d)` 

            =`(3-2d) (3+2d) `......................from (I) 

∴ `(3-2d)(3+2d)=8` 

∴ `9-4d^2=8` 

∴` 4d^2=1       d^2=1/4      d=1/2       or  d=-1/2` 

 Now, `if d=1/2` 

`a+4xx1/2=3`................ from (I)

`a=1 ` 

  If ` d=-1/2` 

`a+4xx(-1/2)=3.......` from (I) 

`a=5` 

∴ the first term of the A. P. is 1 and the common difference is `1/2.` 

or , the first term of the A. P. is 5 and the common difference is `-1/2`.