Question

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

Answer 1

Hint- To find the sum of all two digit odd numbers first we need to find the total numbers of terms. We will calculate the n term of the given series with the help of arithmetic progression. The nth term in the arithmetic progression is given by $a_n=a+(n-1)d$.

Complete step-by-step solution -
Then we will apply the formula of sum of n terms.
All two digit odd positive numbers are $11,13,15,17,........,99$ which are in A.P
With $a=11,d=2,l=99$
Let the number of terms be n.
Therefore the last term is given by
$$\begin{gathered} \Rightarrow a_n=99 \\ \Rightarrow a+(n-1)d=99 \\ \Rightarrow 11+(n-1)\times 2=99 \\ \Rightarrow (n-1)=44 \\ \Rightarrow n=45 \end{gathered}$$
As we know, sum of n terms is given by
$$\begin{gathered} S_n={\displaystyle \frac{n}{2}}(a+l) \\ {S}_n={\displaystyle \frac{45}{2}}(11+99) \\ {S}_n=45\times 55=2475 \end{gathered}$$
Hence, the sum of all two digit odd positive numbers is 2475.

Note- To solve these types of questions, remember the formulas of sum of n terms of an arithmetic series, equation of nth term of arithmetic series. The question can be easily solved by substituting the values in the formula and then finding the unknown term and then substituting the values in the sum formula and after simplifying, sum is calculated.