What is the sum of all two-digit odd positive numbers?

Answer:

2475

Step by Step Explanation:

We know that smallest two digit odd number is 11, and the greatest two digit number is 99.
Since difference between consecutive odd numbers is 2, these numbers form an arithmetic progression with first term = 11, last term = 99 and difference = 2.
If there are total N terms in series, N^th term is given by
T_N = T₁ + (N-1)d
⇒ 99 = 11 + (N-1)(2)
⇒ 2(N - 1) = 99 - 11
⇒ 2(N - 1) = 88
⇒ N - 1 =
88
2

⇒ N = 44 + 1
⇒ N = 45
Now sum of arithmetic progression can be found using standard formula,
S_N = (
N
2
)[2T₁ + (N-1)d]
⇒ S_N = (
45
2
)[2 × 11 + (45-1)(2)]
⇒ S_N = (
45
2
)[22 + 88]
⇒ S_N = (
45
2
)[110]
⇒ S_N = 45 × 55
⇒ S_N = 2475

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