Question

The minimum number of terms of 1+3+5+… that add up to a number exceeding 1357 is?
$\begin{align}
  & \text{a) 15} \\
 & \text{b) 37} \\
 & \text{c) 35} \\
 & \text{d) 17} \\
\end{align}$

Answer 1

Hint: The given sum 1+3+5+… is the sum of a sequence, which is in AP since the difference between consecutive terms is the same. Now we know the sum of n terms of AP is $\dfrac{n}{2}\left[ 2a+(n-1)d \right]$ where a is the first term, d is a common difference, and n is the number of terms.
Hence with the help of this formula and the given condition we can find required n.

Complete step-by-step solution:
Now the given series is $1+3+5+……….$
Here we know the first term is 1.
Difference between any two consecutive terms is \[5-3=3-1=2\]
Now let us try to find the sum of n terms of this series. We know sum of AP is given by $\dfrac{n}{2}\left[ 2a+(n-1)d \right]$
Here we have $a=1$ and $d=2$. Hence sum of n terms will be
$\dfrac{n}{2}\left[ 2(1)+(n-1)(2) \right]$
$=\dfrac{n}{2}\left[ 2+2n-2 \right]$
\[=\dfrac{n}{2}\times \left[ 2n \right]\]
\[={{n}^{2}}\]
Hence now we have the sum $1+3+5+…n$ is \[={{n}^{2}}\]
Now we have to find n such that the sum of the given series is greater than in 1357.
Now let us check the options.
If $n = 37$ then ${{n}^{2}}=37\times 37=1369$
If $n = 36$ then ${{n}^{2}}=36\times 36=1296$
Now for n = 36 we get ${{n}^{2}}=1296<1357$ and for n = 37 we get ${{n}^{2}}=1369$
Hence, the minimum number such that the sum of $1+3+5+…n$ is exceeding 1357 is 1369 and this will happen for $n = 37.$
Hence, the number of terms required is 37.
Option b is the correct option.

Note: Since the sum of odd natural numbers is equal to \[{{n}^{2}}\]we just have to find the square root of sum for thee required n. If the options are not available by hook or crook we can substitute the value of n and find ${{n}^{2}}$ such that it satisfies the required condition. In this case, the condition is to find least n such that${{n}^{2}} >1357$. Hence all we have to do is find n such that ${{n}^{2}}> 1357$ and ${{(n-1)}^{2}}< 1357$.