Question

Find the 100th term of the series 45, 112, 225, 396, 637, .... (This is a cubic series).

Hint:

In cubic series, find the general form using the first few terms and solve for the coefficients.

Answer 1

Step 1: General form of the nth term.
Assume the nth term follows the cubic form: \[ T_n = an^3 + bn^2 + cn + d \] Using the first four terms, we solve for \( a, b, c, d \).
Step 2: Solving the system.
After solving, we get the formula for the nth term: \[ T_n = 7n^3 + 7n^2 + 7n + 31 \]
Step 3: Finding the 100th term.
Substitute \( n = 100 \): \[ T_{100} = 7(100)^3 + 7(100)^2 + 7(100) + 31 = 983,025 \]