Question

If the ratio of $n^{th}$ terms of two A.P.'s is $\left(2n+8\right):\left(5n-3\right)$, then the ratio of the sums of their $n$ terms is

• A. $\left(2n+18\right):\left(5n+3\right)$
• B. $\left(5n-1\right):\left(2n+18\right)$
• C. $\left(2n+18\right):\left(5n-1\right)$
• D. $\left(3n+18\right):\left(4n+1\right)$

Answer 1

The correct option is C $\left(2n+18\right):\left(5n-1\right)$

Given ratio of $n^{th}$ terms of A.P.:

$(2n+8):(5n-3)=\frac{2n-2+2+8}{5n-5+5-3}=\frac{10+(n-1)\times 2}{2+(n-1)\times 5}$

Since $n^{th}$ terms of an A.P. is $t_n=a+(n-1)d$

For the A.P. in numerator, $a_1=10$ and $d_1=2$ and denominator $a_2=2$ and $d_2=5$

Also, Sum of n terms of an A.P. is $S_n=\frac{n}{2}[2a+(n-1\left)d\right]$

Sum of n terms of the A.P. in numerator $=\frac{n}{2}[2\times 10+(n-1)\times 2]$

Sum of n terms of the A.P. in denominator $=\frac{n}{2}[2\times 2+(n-1)\times 5]$

Then, the ratio of the sums of their n nterms is $\frac{\frac{n}{2}[20+2(n-1\left)\right]}{\frac{n}{2}[4+5(n-1\left)\right]}=\frac{2n+18}{5n-1}=(2n+18):(5n-1)$