Finding the remainder when a large sum is divided by a composite number

Calculate the first-order differences between successive terms in the sequence : 14 - 3 = 11 39 - 14 = 25 84 - 39 = 45

Now examine the sequence of differences: 11, 25, 45. Find the second-order differences to reveal the underlying growth pattern: 25 - 11 = 14 45 - 25 = 20

The second-order differences increase by a steady step of 6 (14, 20...). Following this pattern, the next second-order difference should be 20 + 6 = 26. Add this to the last first-order difference to find the next step: 45 + 26 = 71. Calculate the missing term (*): 84 + 71 = 155.

To verify, the next second-order step is 26 + 6 = 32. Next difference = 71 + 32 = 103. Checking the final term: 155 + 103 = 258. This matches the sequence perfectly.

Answer: (b).