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Give the first five terms of the following recursively defined sequence:

a1=2, and ak=ak-1+2 for k2.

Also, give a closed formula for the sequence.

Short Answer

Expert verified

The first five terms of the sequence are: 2,4,6,8,10.

The closed formula for the sequence isak=2k where k is any positive integer.

Step by step solution

01

Step 1. Given Information

A sequence is defined recursively as

a1=2 andak=ak-1+2fork2.

02

Step 2. Find the second and the third term

Substituting 2for kin ak=ak-1+2we get

a2=a2-1+2a2=a1+2a2=2+2a2=4

Again, substituting 3for kin ak=ak-1+2we get

a3=a3-1+2a3=a2+2a3=4+2a3=6

03

Step 3. Find the fourth and the fifth term

In the equation ak=ak-1+2substitute 4for k.

a4=a4-1+2a4=a3+2a4=6+2a4=8

And lastly substitute 5for kin the equation ak=ak-1+2we get

a5=a5-1+2a5=a4+2a5=8+2a5=10

04

Step 4. Find the closed formula

Hence the first five terms of the sequence are 2,4,6,8,10

Clearly, the given sequence is the sequence of consecutive even numbers.

Hence, the sequence can be defined by the closed formula ak=2k, where k is the positive integer.

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