Question

In an AP if ${S_n} = 3{n^2} + 5n$ and ${a_k} = 164$, find the value of k.

Answer 1

Hint – In order to get the right answer to this problem we need to get the value of the first term common difference with the help of the given information and the solve for the value of k. Doing this we will get the right answer.

Complete step-by-step answer:
It has been given that ${S_n} = 3{n^2} + 5n$ and ${a_k} = 164$.
We know that the sum of the first term is the first term itself.
So, we do ${S_1} = 3{(1)^2} + 5(1) = 8$.
Therefore, ${a_1} = a = 8$,
Then the sum of the first two terms is ${S_2} = 3{(2)^2} + 5(2) = 22$.
Therefore, ${a_1} + {a_2} = 22$……(1)
Let the common difference be $d$. So, we can do:
$a + a + d = 22$ (From (1))
On putting the value of $a$ we have calculated above we got the common difference as,
$
  8 + 8 + d = 22 \\
  d = 6 \\
$
Therefore the common difference is 6.
Now we can solve for ${a_k}$. It is given that ${a_k} = 164$ so, we can say that,
${a_k} = a + (k - 1)d = 164$
On putting the value of and d as we have calculated above we got the value of k as,
$
  8 + (k - 1)6 = 164 \\
  8 + 6k - 6 = 164 \\
  6k = 162 \\
  k = \dfrac{{162}}{6} \\
  k = 27 \\
$
The value of k = 27.
Hence the answer to this problem is 27.

Note – To get the right answer to this problem we need to analyze what we can do to get the answer and how we can use the given information in the problem. Once you are done with this your problem will be solved. Like here we have used the general equation of sum to get the values of the first term and common difference and with the help of that we can find the whole series. Knowing this will help you to clear your concept and take you to the right answer.