Question

If the nth term of an arithmetic progression is\[3n + 7\], then what is the sum of its first \[50\] terms?
A) \[3925\]
B) \[4100\]
C) \[4175\]
D) \[8200\]

Answer 1

Hint: in this question we have to find sum of first \[50\] term of given AP series. nth term of AP is given in general form use this to find the first and second term then by using first and second term try to find the common difference of an AP. Then apply the formula of sum of AP to get required value.

Formula Used: In order to calculate sum of n terms of AP formula is given as:
\[{S_n} = \dfrac{n}{2}[2a + (n - 1)d]\]
Where
\[{S_n}\] is sum of of AP
A is first term of AP
n is number of terms in AP
d is common difference of AP

Complete step by step solution: Given: \[3n + 7\]
Now \[{A_n} = 3n + 7\]
First term is \[{A_1} = a = 3 + 7 = 10\]
Second term is \[{A_2} = 3 \times 2 + 7 = 13\]
Common difference is given as \[{A_2} - {A_1}\]
\[d = {A_2} - {A_1}\]
\[d = 13 - 10 = 3\]
Now sum of series is given as
\[{S_n} = \dfrac{n}{2}[2a + (n - 1)d]\]
\[{S_n} = \dfrac{{50}}{2}[2 \times 10 + (50 - 1) \times 3]\]
\[{S_n} = 25[20 + 49 \times 3]\]
\[{S_n} = 4175\]

Option ‘C’ is correct

Note: Here we must remember that we can find any term if general equation of n term of AP is known.
Whenever given series doesn’t follow any pattern then we first try to rearrange the series. If we get any pattern then follow that pattern to get required values. Sometimes by using pattern we are able to find the first term and common ratio or common difference therefore always try to find first term and common ratio or common difference if required. Then apply the formula to get the required value.
Sometime students get confused in between AP and GP the only difference in between them is in AP we talk about common difference whereas in GP we talk about common ratio.