Defining Sequences and Evaluating Terms Examples

Define a sequence where

a_n = the decimal expansion of e to n decimal places.

Find the first four terms of this sequence, starting with n = 1.

Ask your calculator for e or e¹. It probably shows you something like

2.71828183.

To get the n^th term of the sequence, we copy this down to n decimal places. So

a₁ = 2.7

a₂ = 2.71

a₃ = 2.718

a₄ = 2.7182

The first four terms are

2.7, 2.71, 2.718, 2.7182.

Evaluate the first four terms of the sequence defined by a_n = n², starting with n = 1.

To find the 1st term we substitute 1 for n in the formula, and so on:

a₁ = (1)² = 1

a₂ = (2)² = 4

a₃ = (3)² = 9

a₄ = (4)² = 16

The first four terms are 1, 2, 9, 16.