Question

The ratio of the sun of n natural numbers of ap is 7n+1:4n+27.Find the ratio of their 11 term

Answer 1

We may have an important formula,

$\boxed{S_n=n\left(a_{\frac{n+1}{2}}\right)}$

This formula means that the sum of a first few terms of an AP is the product of the no. of terms involved in the sum and the middle term among them, if there are odd no. of terms.

This is just the concept that sum equals average multiplied by no. of terms.

In the formula $a_{\frac{n+1}{2}}$ is the middle term. According to the question, this is the 11th term.

$a_{\frac{n+1}{2}}=a_{11}\\\\\\\dfrac{n+1}{2}=11\\\\\\n=21$

So we may consider the sum of the first 21 terms of the two APs.

$S_{1(n)}:S_{2(n)}=7n+1:4n+27\\\\\\S_{1(21)}:S_{2(21)}=7\cdot21+1:4\cdot21+27\\\\\\21\cdot a_{1(11)}:21\cdot a_{2(11)}=148:111\\\\\\\mathbf{a_{1(11)}:a_{2(11)}=4:3}$

So we had a new idea!

The ratio of first $n$ terms of two APs is equal to the ratio of $\left(\dfrac{n+1}{2}\right)^{th}$ term of the APs, i.e.,

$S_{1(n)}:S_{2(n)}=a_{1\left(\frac{n+1}{2}\right)}:a_{2\left(\frac{n+1}{2}\right)}$