Question

The sum of n terms of an A.P. is \[3{n^2} + 5n\] and its \[{m^{th}}\]term is 164, find the value of m.

Answer 1

Hint:General term (\[{n^{th}}\]term) of an A.P. is
 \[{a_n} = a + (n - 1)d\]
Where a =1st term & d =common difference
Sum of first n terms of an AP
\[{S_n} = \dfrac{n}{2}[2a + (n - 1)d]\]
Where a =1st term & d =common difference
Or,
\[{S_n} = \dfrac{n}{2}[a + {a_n}]\]
If there are only n terms in an AP, then \[{a_n}\]=l, l=last term
We will put n=1, 2 in the sum formula and evaluate the first and second term of the A.P.
Then we can easily derive the common difference. By putting all these in \[{m^{th}}\]term we can find the value of m.

Complete step-by-step answer:
Let, \[{a_1},{a_2},{a_3},.........,{a_n}\]be the n terms of the given A.P.
Given that, the sum of n terms of an A.P. is \[3{n^2} + 5n\].
That is \[{S_n} = 3{n^2} + 5n\]…. (1)
Let us put n=1 we get,
\[{S_1} = 3 \times {1^2} + 5 \times 1 = 3 + 5 = 8\]
We know that the sum of first term =first term
That is \[{a_1} = 8\]
Let us put n=2 in (1) we get,
\[{S_2} = 3 \times {2^2} + 5 \times 2 = 3 \times 4 + 10 = 12 + 10 = 22\]
Also we know that sum of first 2 terms =First term +second term =22
Also we know that \[{a_1} = 8\]
That is \[8 + {a_2} = 22\]
\[{a_2} = 22 - 8 = 14\]
Thus, \[{a_1} = 8\] and \[{a_2} = 14\]
We know that the common difference d = second term –first term
That is \[d = {a_2} - {a_1}\]
Let us substitute the known values, then we get
\[d = 14 - 8\]
\[d = 6\]
Now, \[{a_1} = 8\] and \[d = 6\]
Again, it is given that, \[{m^{th}}\]term of the A.P. is 164
We know that the \[{n^{th}}\]term of an A.P. is \[{a_n} = a + (n - 1)d\]
Where a =1st term & d =common difference
Let us substitute \[n = m\] in\[{a_n}\], we get
\[{a_m} = a + (m - 1)d\]
We have, \[a = 8,d = 6\] and \[{a_m}\]=164
By substituting the values we get,
\[{a_m} = 8 + (m - 1) \times 6 = 164\]
By solving on the left hand side of the equation we get,
\[8 + 6m - 6 = 164\]
\[6m = 164 - 2\]
Let us divide by six on both sides then we get,
\[m = \dfrac{{162}}{6}\]
\[m = 27\]

Hence the value of m is 27.

Note: If all the terms of a progression except the first one exceeds the preceding term by a fixed number, then the progression is called arithmetic progression. If “a” is the first term of a finite AP and d is a common difference, then AP is written as\[a,{\rm{ }}a + d,{\rm{ }}a + 2d \ldots a + \left( {n - 1} \right)d\] .