Question

The sum of n terms in two AP's are in the ratio 5n +1:n + 6, then the ratio of their 5th terms is​

Answer 1

The ratio of their 5th term is 46:15.

Given,

The sum of n terms in two APs is in the ratio (5n +1):(n + 6).

To Find,

The ratio of their 5th term.

Solution,

The sum of n terms of A.P are in ratio (5n +1):(n + 6)

So,

(2a+(n-1)d)/(2a'+(n-1)d') = (5n +1):(n + 6)

diving the numerator and denominator of L.H.S by 2

(a+(n-1)d/2)/(a'+(n-1)d'/2) = (5n +1):(n + 6)

The fifth terms are a+4d=a+(9-1)d/2 and a'+4d'=a'+(9-1)d'/2

Now, substituting the values

(5n +1):(n + 6) =  5x9+1 : 9+6 = 46:15

Hence, the ratio of their 5th term is 46:15.

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Answer 2

Answer:

Thus, the ratio of both the AP's 5th terms is 46:15.

Step-by-step explanation:

Given - The ratio of the sum of n terms in 2 APs - $(5n+1):(n+6)$

To find - The ratio of the 5th terms in both APs

Solution -
We are given the ratio of the sum of n terms of both APs to be $(5n+1):(n+6)$

So, we can write this as $\frac{2a+(n-1)d}{2a'+(n-1)d'} = \frac{5n +1}{n + 6}$

This can further be simplified as follows -

$\frac{\frac{a+(n-1)d}{2} }{\frac{a'+(n-1)d'}{2} } = \frac{5n+1}{n+6}$

Thus, we can write the respective 5th terms to be -

$a+4d = \frac{a+(9-1)d}{2} \\\\a'+4d' = \frac{a'+(9-1)d'}{2}$

Substituting the values in these 2 equations, we get
$\frac{5n+1}{n+6} = \frac{5(9) + 1}{9+6} = \frac{46}{15}$

Thus, we can conclude that the ratio of their 5th terms is 46:15.

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