Question

After expressing ${18}^3$ as the 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 = $\frac{{18}^3}{18}$ = ${18}^2$ = $324$, 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,

${18}^3$ = 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.

Sum of first and last terms will be 307 + 341 = 648