Question

13. Find the sum of all two-digit odd positive numbers.

Answer 1

Step 1: Note the given data

Given sequence are all two-digit odd positive numbers.

First-term $t_1=a=11$

First-term $t_2=13$

Last-term $t_n=99$

Given sequence is in A.P.

Step 2: Finding the common difference in A.P.

The formula for finding the common difference in A.P. $={(n+1)}^{th}term-n^{th}term$

Common difference $=13-11$

$=2$

Step 3: Finding the $n^{th}$ term of A.P.

The formula for finding $n^{th}$ term of A.P. $a+(n-1)d$

$n^{th}$ term of A.P. $t_n=11+(n-1)\times 2$

$$\begin{gathered} =11+2n-2 \\ =9+2n \end{gathered}$$

Step 4: Equating both the $n^{th}$ term

$9+2n=99$

$\Rightarrow 2n=99-9$

$\Rightarrow 2n=90$

$\Rightarrow n=\frac{90}{2}$

$\Rightarrow n=45$

Step 5: Finding the sum of A.P.

The formula of the sum of first $n$terms of the AP is $=\frac{n}{2}(t_1+t_n)$

The sum

$$\begin{gathered} =\frac{45}{2}(11+99) \\ =\frac{45}{2}\times 110 \\ =45\times 55 \\ =2475 \end{gathered}$$

Hence, the sum of the given A.P. is $2475$.