# Find the sum of all odd natural numbers from 9 to 99, inclusive.

June 26, 2021 | education

| A sequence of odd natural numbers from 9 to 99 inclusive is an arithmetic progression an with the first term a1 equal to 9 and the difference d equal to 2.

Let’s find the number of the last member of the given progression.

To do this, solve the following equation:

9 + (n – 1) * 2 = 99;

9 + 2n – 2 = 99;

7 + 2n = 99;

2n = 99 – 7;

2n = 92;

n = 92/2;

n = 46.

Therefore, there are 46 members in this sequence.

We find the sum of the first 46 terms of this arithmetic progression:

S46 = (2 * a1 + d * (46 – 1)) * 46/2 = (2 * a1 + d * 45) * 23 = (2 * 9 + 2 * 45) * 23 = (18 + 90) * 23 = 108 * 23 = 2484.

Answer: the sum of these numbers is 2484.

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