Question

10+12+14+...+(2n)=90 n=?

Answer 1

n=10

Explanation:

The given equation is

`10+12+14+...+2n=90` ...(1)

Let nth term is 2n.

We need to find the values of n.

It is clear that
`S_n=10+12+14+...+2n` is sum of A.P., whose first term is 10 and common difference is 2.

Sum of A.P. is

`S_n=(n)/(2)[2a+(n-1)d]`

where, a is first term and d is common difference.

Substitute a=10 and d=2 in the above formula.

`S_n=(n)/(2)[2(10)+(n-1)2]`

`10+12+14+...+2n=n[10+n-1]`

`10+12+14+...+2n=n[9+n]` ...(2)

From (1) and (2), we get

`n(9+n)=90`

`n^2+9n-90=0`

Splitting middle term, we get

`n^2+15n-6n-90=0`

`n(n+15)-6(n+15)=0`

`(n+15)(n-6)=0`

`n=-15,6`

Since, n represents the number of terms so n cannot be a negative number, therefore number of term is 6.

Note: nth term and variable n both are different.

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

`a_6=10+(6-1)2=10+10=20`

Sixth term is 20. So,

`2n=20`

`n=10`

Therefore, the value of n is 10.