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

Hint: 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.