Find the next term in the series given below : 2, 10, 30, 68, 130, ?

Concept: A "find the next term" number series is settled by fitting a rule to the position index n = 1, 2, 3, ..., not by guessing. Two standard routes are available. One is to find a closed form f(n) that reproduces every given term, after which the missing term is simply f(n) evaluated at the next index. The other is to build the table of successive differences: when some row of differences becomes constant, the series is a polynomial in n, and that table can be extended one column further to recover the next term. Agreement between the two routes is the usual check.

Applying both routes to 2, 10, 30, 68, 130:

1. Index the terms by position: n = 1 gives 2, n = 2 gives 10, n = 3 gives 30, n = 4 gives 68 and n = 5 gives 130.

2. Write down the cube of each position: 13 = 1, 23 = 8, 33 = 27, 43 = 64 and 53 = 125.

3. Subtract each cube from the term at the same position: 2 - 1 = 1, 10 - 8 = 2, 30 - 27 = 3, 68 - 64 = 4 and 130 - 125 = 5. The leftover is exactly the position n every time.

4. The closed form is therefore f(n) = n3 + n, and it reproduces all five given terms.

5. The term asked for sits at position n = 6, so f(6) = 63 + 6 = 216 + 6 = 222.

Cross-check with the table of successive differences:

1. First differences: 10 - 2 = 8, 30 - 10 = 20, 68 - 30 = 38 and 130 - 68 = 62.

2. Second differences: 20 - 8 = 12, 38 - 20 = 18 and 62 - 38 = 24.

3. Third differences: 18 - 12 = 6 and 24 - 18 = 6. This row is constant, so the series is a cubic in n, which agrees with f(n) = n3 + n.

4. Extend the rows: the next third difference is again 6, so the next second difference is 24 + 6 = 30 and the next first difference is 62 + 30 = 92.

5. The next term is therefore 130 + 92 = 222, the same value the closed form gives.

Two slips worth avoiding:

• Treating the first differences 8, 20, 38, 62 as though they grew by a constant amount.

• Stopping at n3 without noticing the extra + n that the leftovers 1, 2, 3, 4, 5 point to.

The next term of the series is 222.