Question

Sequences:

See the following attached image, where you can see sets of circles, which follow a certain pattern:
Number of circles in next row 6: 21 circles, and row 7: 28 circles

1) Determine the number of circles in the nth pile (general term)

2) What is the number of circles when n tends to infinity?

Answer 1

The sequence is

• 1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, ...

Each term is the sum of the consecutive numbers from 1 to that number.

The nth term is the sum of the first n numbers:

• aₙ = 1 + 2 + 3 + ... + n
• aₙ = 1/2n(1 + n) (formula for sum of the n terms of arithmetic progression with the first term of 1 and common difference of 1)
• aₙ = n(n + 1)/2

1) The number of circles in the nth pile is n(n + 1)/2

2) When n tends to infinity the number of circles tends to infinity