Question

If $\dfrac{{3 + 5 + 7 + ...\,n\,terms}}{{5 + 8 + 11 + ...\,10\,terms\,}} = 7$, then the value of $n$ is:
A. $35$
B. $36$
C. $37$
D. $40$

Answer 1

Hint: The given problem requires us to find the value of n when the value of a rational expression is given to us involving two series or progressions. So, we first find out the value of numerator and denominator by using the formula of summation of an arithmetic progression. For finding out the sum of an arithmetic progression, we need to know the first term, the common difference and the number of terms in the arithmetic progression. We can find out the common difference of an arithmetic progression by knowing the difference of any two consecutive terms of the series.

Complete step by step answer:
So, we have, $\dfrac{{3 + 5 + 7 + ...\,n\,terms}}{{5 + 8 + 11 + ...\,10\,terms\,}} = 7$
The numerator and denominator of the rational expression consists of arithmetic progressions. So, we first calculate the numerator and denominator separately by using the summation formula of AP. So, $3 + 5 + 7 + ...\,n\,terms$. Here, we have the first term $ = a = 3$. So, Common difference of AP $ = 5 - 3 = 2$.

Now, we can find the sum of the given arithmetic progression using the formula $S = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$.
So, we have, $3 + 5 + 7 + ...\,n\,terms = \dfrac{n}{2}\left[ {2 \times 3 + 2\left( {n - 1} \right)} \right]$
Cancelling the common factors in numerator and denominator, we get,
$ \Rightarrow 3 + 5 + 7 + ...\,n\,terms = n\left[ {3 + \left( {n - 1} \right)} \right]$
Simplifying the expression,
$ \Rightarrow 3 + 5 + 7 + ...\,n\,terms = n\left( {n + 2} \right)$
Also, we have, $5 + 8 + 11 + ...\,10\,terms$. Here, first term $ = a = 5$
Common difference of AP $ = 8 - 5 = 3$.

Now, we can find the sum of the given arithmetic progression using the formula $S = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$.
So, we have, $5 + 8 + 11 + ...\,10\,terms = \dfrac{{10}}{2}\left[ {2\left( 5 \right) + \left( {10 - 1} \right)\left( 3 \right)} \right]$
$ \Rightarrow 5 + 8 + 11 + ...\,10\,terms = 5\left[ {10 + 9\left( 3 \right)} \right]$
Simplifying the expression,
$ \Rightarrow 5 + 8 + 11 + ...\,10\,terms = 5 \times 37$
$ \Rightarrow 5 + 8 + 11 + ...\,10\,terms = 185$
So, we get the rational expression as $\dfrac{{3 + 5 + 7 + ...\,n\,terms}}{{5 + 8 + 11 + ...\,10\,terms\,}} = \dfrac{{n\left( {n + 2} \right)}}{{185}} = 7$
Now, cross multiplying the terms of the equation and opening the brackets, we get,
$ \Rightarrow {n^2} + 2n = 1295$

Now, we solve the above quadratic equation using the splitting the middle term method to find the value of n.
$ \Rightarrow {n^2} + 37n - 35n - 1295 = 0$
We split the middle term $2n$ into two terms $37n$ and $ - 35n$ since the product of these terms, $ - 1295{n^2}$ is equal to the product of the constant term and coefficient of ${x^2}$ and sum of these terms gives us the original middle term, $2n$. Now, taking out the common factors from the terms, we get,
$ \Rightarrow n\left( {n + 37} \right) - 35\left( {n + 37} \right) = 0$
$ \Rightarrow \left( {n - 35} \right)\left( {n + 37} \right) = 0$
Now, either $n - 35 = 0$ or $n + 37 = 0$.
So, we get, $n = 35$ or $n = - 37$.
Now, $n$ cannot be negative since it denotes the number of terms. So, we get, $n = 35$.

Hence, option A is the correct answer.

Note: The sum of $n$ terms of an arithmetic progression can be calculated if we know the first term, the number of terms and difference of the arithmetic series as: $S = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$. Splitting the middle term can be a tedious process at times when the product of the constant term and coefficient of ${x^2}$ is a large number with a large number of divisors. Special care should be taken in such cases.