Question

How do you find the sum of counting numbers from 1 to 25 inclusive?

Answer 1

Hint: In this problem, we have to find the sum of counting numbers from 1 to 25 inclusive. We can first write the series from 1 to 25 in ascending and descending order. We can then pair the term in ascending and descending order and obtain their sum. We can multiply the number of pairs and the sum of the pairs and we can divide it by 2 as we have doubled the sum required to get the answer.

Complete step by step answer:
We know that we have to find the sum of counting numbers from 1 to 25 inclusive.
We can write the series from 1 to 25 in ascending and descending order.
  Ascending order $$\Rightarrow 1+2+3+4+5+........+21+22+23+24+25$$
Descending order$$\Rightarrow 25+24+23+22+21+........+5+4+3+2+1$$
We can see that, if we pair it off and add, we get the same sum, i.e. 26.
We can also see that there are 25 pairs, but we have doubled the sum required, so we can add the number of pairs and sum of the pairs and divide it by 2, we get
$$\Rightarrow Sum={\displaystyle \frac{25\times 26}{2}}=325$$

Therefore, the sum of counting numbers from 1 to 25 inclusive is 325.

Note: We can also find the sum from the arithmetic progression.
We know that the sum of series formula is,
$$S={\displaystyle \frac{n}{2}}(a+T_n)$$ …… (1)
We know that the given series is,
$$\Rightarrow 1+2+3+4+5+........+21+22+23+24+25$$
Here, first term, a = 1, common difference, d = 1 and number of terms, n = 25.
We can substitute the above values in (1), we get
$$\begin{array}{rl}& \Rightarrow S={\displaystyle \frac{25}{2}}(25+1)={\displaystyle \frac{650}{2}}\\ & \Rightarrow S=325\end{array}$$