Question

Question 6
The ratio of the ${11}^{th}$ term to the ${18}^{th}$ term of an AP is 2:3. Find the ratio of the $5^{th}$ term to the ${21}^{st}$ term and also the ratio of the sum of the first five terms to the sum of the first 21 terms.

Answer 1

Let $a$ and $d$ be the first term and common difference of an AP.
Given that, $a_{11}:a_{18}=2:3$
$\Rightarrow \frac{a+10d}{a+17d}=\frac{2}{3}$
$\Rightarrow 3a+30d=2a+34d$
$\Rightarrow a=4d$ ...(i)
Now, $a_5$ = a + 4d = 4d + 4d = 8d [from Eq.(i)]
And $a_{21}$ = a + 20d = 4d + 20d = 24d [from Eq. (i)]
$a_5:a_{21}$ = 8d : 24d = 1 : 3
Now, sum of the first five terms, $S_5=\frac{5}{2}[2a+(5-1\left)d\right]$
$=\frac{5}{2}\left[2\right(4d)+4d][fromEq.(i\left)\right]$
$=\frac{5}{2}(8d+4d)=\frac{5}{2}\times 12d=30d$
And, sum of the first 21 terms, $S_{21}=\frac{21}{2}[2a+(21-1\left)d\right]$
$=\frac{21}{2}\left[2\right(4d)+20d]=\frac{21}{2}\times 28d=294d$ from Eq..(i)]
So, ratio of the sum of the first five terms to the sum of the first 21 terms is,
$S_5:S_{21}=30d:294d=5:49$