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Question
Find the sum of odd natural numbers from 1 to 101.
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Solution
The odd natural numbers from 1 to 101 are
1, 3, 5,....., 101
The above sequence is an A.P.
∴ a = 1, d = 3 – 1 = 2
Let the number of terms in the A.P. be n.
Then, tn = 101
Since tn = a + (n – 1)d,
101 = 1 + (n – 1)(2)
∴ 101 = 1 + 2n – 2
∴ 101 = 2n – 1
∴ 102 = 2n
∴ n = `102/2`
∴ n = 51
Now, `S_n = n/2 (t_1 + t_n)`
∴ `S_51 = 51/2 (1 + 101)`
= `51/2 (102)`
= 51 × 51
= 2601
∴ The sum of odd natural numbers from 1 to 101 is 2601.
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