# Finding the sum of arithmetic series when last term and common difference is given .

The last term of an arithmetic series of 20 terms is 195 and common difference is 5. Calculate the sum of the series.

• Your thoughts?? – Rohan Dec 15 '17 at 13:54

The $n$th term of an arithmetic series is given by $$T_n=a+(n-1)d$$where $T_n$ is the $n$th term. If you know $n$, the last term and $d$ you can use this to calculate a value for $a$, the first number. From here you can you the summation formula for an arithmetic series that is $$S_n=\frac{n}{2}\big(2a+(n-1)d\big)$$ where $S_n$ is the sum of the series.
The $nth$ term of an AP is given by $T_n=a+(n-1)d$ where $a$ is the first term and $d$ is the common difference.
Given $a + (20-1)5 = 195$ $\implies a = 100$
The sum of first $n$ terms of an AP is $S_n = \frac{n}{2}(2a + (n-1)d)$
Therefore, $S_{20} = \frac{20}{2}(2\cdot100+(20-1)\cdot5) = 2950$
• The sentence, "Therefore the sum of the first $n$ terms of an AP is ..." should not include the word therefore. Also, you solved for $S_{20}$ without specifying that you set $n = 20$. – N. F. Taussig Dec 16 '17 at 11:08
We know that the formula for an arithmetic series is $$S_n=\frac{n}{2}(a_1+a_n)$$ We already know $a_n$ and $n$, therefore we only have to figure out $a_1$. Is there a way you can use the fact that the common difference is 5 and $n=20$ to figure out $a_1$? (Perhaps, subtract 5 from 195 a specific number of times to reach $a_1$).