Question

If nth terms of two A.P’s are $3n + 8$ and $7n + 15$, then the ratio of their $12th$ terms will be

Answer 1

As the nth terms of both the AP’s are given to us so it will be easy for us to find the value of $12th$ terms of both the AP’s. We just have to put $n = 12$ in both the nth terms of AP’s and on solving further we will get the values of both the $12th$ terms. Then divide the value of $12th$ term of first AP by the value of $12th$ term of second AP to get the ratio.

Complete step by step solution:
In the question the nth terms of both the AP’s are given to us. So, let the nth term of first AP is
${a_n} = 3n + 8$ -------- (i)
And the nth term of the second AP is
${a_n} = 7n + 15$ -------- (ii)
So, let’s first find the $12th$ term of the first AP. So, put $n = 12$ in the equation (i)
$\Rightarrow {a_{12}} = 3\left( {12} \right) + 8$
On multiplying $3$ by $12$ we get
$\Rightarrow {a_{12}} = 36 + 8$
By doing addition the above equation becomes
$\Rightarrow {a_{12}} = 44$ ------------- (iii)
Now we will find the $12th$ of the second AP. For this again put $n = 12$ in the equation (ii). By doing this the equation (ii) becomes
$\Rightarrow {a_{12}} = 7\left( {12} \right) + 15$
By doing multiplication we get
$\Rightarrow {a_{12}} = 84 + 15$
On adding both the terms we get
$\Rightarrow {a_{12}} = 99$ ----------- (iv)
Now to find the ratio of $12th$ terms of both the AP’s, divide equation (iii) by equation (iv)
$\dfrac{{(iii)}}{{(iv)}} \Rightarrow \dfrac{{44}}{{99}}$
Now clearly both the terms in the numerator and the denominator can be divisible by the number $11$ .Therefore,
$\dfrac{{(iii)}}{{(iv)}} \Rightarrow \dfrac{4}{9}$
Hence, the ratio of their $12th$ terms is $\dfrac{4}{9}$.

Note:
There is another method also to find the $12th$ terms of both the AP’s. As we know, the nth term in the AP is of the form ${a_n} = a + \left( {n - 1} \right)d$ where a is the first term of AP and d is the difference between two terms. If we put $n = 12$ then our $12th$ is of the form ${a_{12}} = a + 11d$ .So, to find the value of $12th$ terms we have to first find the first term of both the AP’s by putting n is equal to one and we have to also find the difference. And then we will be able to find the ratio of both the terms.