Question

After expressing ${18}^3$ as a sum of consecutive odd numbers, find the sum of the first and last terms.

Answer 1

Step 1: Find two middle numbers, as 18 is an even number

• Middle number = $\left({18}^3\right)/18={18}^2=324$, which is an even number

• So, $323$ and $325$ are the middle numbers in this sequence of odd numbers.

Step 2: Write ${18}^3$ as the sum of consecutive odd numbers with the help of $323$ and $325$.

• Since we need 18 such odd numbers, we find 8 consecutive odd numbers to the left and right of $323$ and $325$ respectively.

Hence,

${14}^3=307+309+311+313+315+317+319$
$+321+323+325+327+329+331+333$
$+335+337+339+341$

• The first and last terms are $307$ and $341$.

• So, the sum of the first and last term $=307+341=648$