Question

Find the sum of all odd natural numbers from 1 to 150.

Answer 1

Odd numbers from 1 to 150 are 1, 3, 5, 7,…, 149.
Now, we observe that the numbers are in an A.P.
Here,
First term, a = 1
Common difference, d = t₂ – t₁ = 3 – 1 = 2
Let the number of these odd numbers be n.
We have: t_n = 149
We know that the n^th term of an A.P. is t_n = a + (n – 1)d.
Thus, we get:
149 = 1 + (n – 1)2
$\Rightarrow$149 – 1 = 2(n – 1)
$\Rightarrow$148 = 2(n – 1)
$\Rightarrow$(n – 1) = 74
$\Rightarrow$n = 74 + 1 = 75
There are 75 odd numbers from 1 to 150.
Now,
We know that the sum of n terms of an A.P. is $S_n=\frac{n}{2}\left[2a+(n-1)d\right]$.
To find the sum of these odd numbers, we need to take a = 1, d = 2 and n = 75.
Thus, we have:
$$\begin{gathered} S_{75}=\frac{75}{2}[2\times 1+(75-1\left)2\right]=\frac{75}{2}[2+148] \\ =\frac{75}{2}\times 150 \\ =5625 \end{gathered}$$