The two digit odd positive integers are
11, 13, 15,…, 99
a2 – a1 = 13 – 11 = 2
a3 – a2 = 15 – 13 = 2
∵ a3 – a2 = a2 – a1 = 2
Therefore, the series is in AP
Here, a = 11, d = 2 and an = 99
We know that,
an = a + (n – 1)d
⇒ 99 = 11 + (n – 1)2
⇒ 99 – 11 = (n – 1)2
⇒ 88 = (n – 1)2
⇒ 44 = (n – 1)
⇒ n = 45
Now, we have to find the sum of this AP
⇒ S45 = 45[11 + 44]
⇒ S45 = 45[55]
⇒ S45 = 2475
Hence, the sum of all two digit odd numbers are 2475.