Question

An arithmetic series contains 20 numbers. The first number is 102. The last number is 159. Which expression represents the sum of the series?

Answer 1

Answer: The expression is,

$\frac{20}{2}(102+159)$

Step-by-step explanation:

Since, the sum of an A.P. is,

$S_n=\frac{n}{2}(a+a_n)$

Where, n is the number of terms,

a is the first term of the A.P.,

$a_n$ is the last term of the A.P.,

Here, we have given,

a = 102, $a_n$ = 159, and n = 20,

Thus, the sum of the given A.P. is,

$S_20=\frac{20}{2}(102+159)$

Which is the required expression.

Answer 2

If $a_1=102$ is the first term in the series, then the next term is $a_2=a_1+k$ for some fixed $k$, the term after that is $a_3=a_2+k=a_1+2k$, and so on. So the 20th term in the series would be $a_{20}=a_1+(20-1)k=102+19k=159$.

We have enough info to find $k$:

$102+19k=159\implies19k=57\implies k=3$

So the sum of the series is given by

$\displaystyle\sum_{n=1}^{20}(a_1+3(n-1))=\sum_{n=1}^{20}(3n+99)=2610$