Question

Consider the AP’s 17, 21, 25 and 16, 21, 26. Find the sum of 100 common terms appearing in the two series.

Answer 1

Hint: In this problem, we have to find the sum of 100 common terms appearing in the given two series which are 17, 21, 25 and 16, 21, 26. We can first find the common term from the given two series which will be the first term. We can then find the common difference d, by taking LCM of the given two series. We have to find the sum of 100 common term, then n will be 100, we can substitute these values in the formula \[\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)\] to get the sum of 100 common terms.

Complete step-by-step solution:
Here, we have to find the sum of 100 common terms appearing in the given two series which are 17, 21, 25 and 16, 21, 26.
We can see that, we have 21 as the common term in the given two series, which will be the first term
\[\Rightarrow a=21\]
We can see that we have 4 as the common difference for 17, 21, 25 and 5 as the common difference for 16, 21, 26. We can now take LCM for the both common difference, we get
LCM of 4 and 5 is 20.
\[\Rightarrow d=20\]
We are already given that,
\[\Rightarrow n=100\]
We can now substitute the above values in the formula \[\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)\], we get
\[\begin{align}
  & \Rightarrow Sum=\dfrac{100}{2}\left( 2\left( 21 \right)+\left( 100-1 \right)20 \right) \\
 & \Rightarrow Sum=50\left( 42+1980 \right)=101100 \\
\end{align}\]
Therefore, the sum of 100 common terms appearing in the given two series 17, 21, 25 and 16, 21, 26 is 101100.

Note: We should always remember that, the formula two find the sum using the number of terms, n, first term, a and the common difference, d is \[\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)\]. We should also know that, we have two common differences, then we can take LCM of the given terms, to get the common difference, which will be common for both the series.