Expert answers to "Find the ﬁrst four terms of each of..."

2alr8w

Answered question

2022-01-23

Find the ﬁrst four terms of each of the recursively deﬁned sequences

d_{k} = k {(d_{k - 1})}^{2}

for all integers

k \geq 1

d_{0} = 3

ma90t66690

Beginner2022-01-24Added 7 answers

The recursive definition of the given sequence is

d_{k} = k {(d_{k - 1})}^{2}

for all integers

k \geq 1

The first term of the sequence is

d_{0} = 3

Evaluate the second term

d_{1}

using

d_{k} = k {(d_{k - 1})}^{2}

and

d_{0} = 3

d_{1} = (1) {(d_{1 - 1})}^{2}

= {(d_{0})}^{2}

= 3^{2}

= 9

Now, evaluate the third term

d_{2}

using

d_{k} = k {(d_{k - 1})}^{2}

and

d_{1} = 9

d_{2} = (2) {(d_{2 - 1})}^{2}

= 2 {(d_{1})}^{2}

= 2 {(9)}^{2}

= 162

Now, evaluate the fourth term

d_{3}

using

d_{k} = k {(d_{k - 1})}^{2}

and

d_{2} = 162

d_{3} = (3) {(d_{3 - 1})}^{2}

= 3 {(d_{2})}^{2}

= 3 {(162)}^{2}

= 78732

Thus, the first four terms are:

3, 9, 162, 78732

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