Question

Find five numbers in A.P. whose sum is `12 1/2` and the ratio of the first to the last terms is 2 : 3.

Answer 1

Let the five numbers in A.P. be (a – 2d), (a – d), a, (a + d) and (a + 2d).

Then, `(a - 2d) + (a - d) + a + (a + d) + (a + 2d) = 12 1/2`

`\implies 5a = 25/2`

`\implies a = 5/2`

It is given that 

`(a - 2d)/(a + 2d) = 2/3`

`\implies` 3a – 6d = 2a + 4d

`\implies` a = 10d

`\implies 5/2=10 d` 

`\implies d = 1/4`

`\implies a = 5/2` and `d = 1/4` 

Thus, we have 

`a - 2d = 5/2 - 2 xx 1/4`

= `5/2 - 1/2`

= `4/2`

= 2

`a - d = 5/2 - 1/4`

= `(10 - 1)/4`

= `9/4`

`a = 5/2`

`a + d = 5/2 + 1/4`

= `(10 + 1)/4`

= `11/4`

`a + 2d = 5/2 + 2 xx 1/4`

= `5/2 + 1/2`

= `6/2`

= 3

Thus, the five numbers in A.P. = `2, 9/4, 5/2, 11/4` and 3

= 2, 2.25, 2.5, 2.75 and 3