Question

Find the sum \[7 + 10\frac{1}{2} + 14 + . . . + 84\]

 

Answer 1

(v) \[7 + 10\frac{1}{2} + 14 + . . . + 84\]

Common difference of the A.P is

(d) =

`=10 1/2-7`

`=21/2 - 7`

`=(21-14)/2`

`=7/2`

So here,

First term (a) = 7

Last term (l) = 84

Common difference (d) = `7/2`

So, here the first step is to find the total number of terms. Let us take the number of terms as n.

Now, as we know,

`a_n = a + (n-1) d`

So, for the last term,

`84 = 7 + (n-1) 7/2`

`84 = 7 + (7n)/2 - 7/2`

`84 = (14-7)/2 + (7n)/2`

84 (2) = 7 + 7n

Further solving for n,

7n = 168 - 7

`n = 161/7`

n = 23

Now, using the formula for the sum of n terms, we get

`S_n = 23/2 [2(7) + (23-1) 7/2]`

    ` = 23/2 [14+(22)7/2]`

   `=23/2(14+77) `

  `= 23/2 (91)`

On further simplification, we get,

`S_n = 2093/2`

Therefore, the sum of the A.P is `S_n = 2093/2`