# The ratio of the 11th term to the 18th

Question:

The ratio of the 11th term to the 18th term of an AP is 2 : 3. Find the ratio of the 5th term to the 21st term and also the ratio of the sum of the first five

terms to the sum of the first 21 terms.

Solution:

Let a arid d be the first term and common difference of an AP

Given that, $a_{11}: a_{18}=2: 3$

$\Rightarrow$ $\frac{a+10 d}{a+17 d}=\frac{2}{3}$

$\Rightarrow$ $3 a+30 d=2 a+34 d$

$\Rightarrow$  $a=4 d$ $\ldots$ (i)

Now, $a_{5}=a+4 d=4 d+4 d=8 d$   [from Eq. (i)]

and $a_{n+1}=a+20 d=4 d+20 d=24 d$  [from Eq. (i)]

$\therefore \quad a_{5}: a_{21}=8 d: 24 d=1: 3$

Now, sum of the first five terms, $S_{5}=\frac{5}{2}[2 a+(5-1) d]$

$=\frac{5}{2}[2(4 d)+4 d]$ [from Eq. (i)]

$=\frac{5}{2}(8 d+4 d)=\frac{5}{2} \times 12 d=30 d$

and sum of the first 21 terms, $S_{21}=\frac{21}{2}[2 a+(21-1) d]$

$=\frac{21}{2}[2(4 d)+20 d]$ [from Eq. (i)]

$=\frac{21}{2}(28 d)=294 d$

So, ratio of the sum of the first five terms to the sum of the first 21 terms

S5 :S21 = 30 d :294 d = 5:49

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