Chapter 7: Q. 21 (page 592)

The Fibonacci numbers may be computed with the formula
$f_{k} = \frac{{(1 + \sqrt{5})}^{k} - {(1 - \sqrt{5})}^{k}}{2^{k} \sqrt{5}}$
Use this formula to compute f $f_{1}, f_{2}, f_{3}, f_{4}, and f_{5} .$ (Imagine computing $f_{100}$ .)

Short Answer

Expert verified

The first five terms of the sequence are $f_{1} = 1, f_{2} = 1, f_{3} = 2, f_{4} = 3, & f_{5} = 5 .$

Step by step solution

Step 1. Given information.

The given sequence is the following.

$f_{k} = \frac{{(1 + \sqrt{5})}^{k} - {(1 - \sqrt{5})}^{k}}{2^{k} \sqrt{5}}$

Step 2. The first term of the sequence.

Substitute $k = 1$ in the sequence for the first term.

$f_{k} = \frac{{(1 + \sqrt{5})}^{k} - {(1 - \sqrt{5})}^{k}}{2^{k} \sqrt{5}} f_{1} = \frac{{(1 + \sqrt{5})}^{1} - {(1 - \sqrt{5})}^{1}}{2^{1} \sqrt{5}} f_{1} = \frac{2 \sqrt{5}}{2 \sqrt{5}} f_{1} = 1$

So the first term of the sequence is $f_{1} = 1 .$

Step 3. The second term of the sequence.

Substitute $k = 2$ in the sequence for the second term.

$f_{k} = \frac{{(1 + \sqrt{5})}^{k} - {(1 - \sqrt{5})}^{k}}{2^{k} \sqrt{5}} f_{2} = \frac{{(1 + \sqrt{5})}^{2} - {(1 - \sqrt{5})}^{2}}{2^{2} \sqrt{5}} f_{2} = \frac{4 \sqrt{5}}{4 \sqrt{5}} f_{2} = 1$

So the second term of the sequence is $f_{2} = 1 .$

Step 4. The third term of the sequence.

Substitute $k = 3$ in the sequence for the third term.

$f_{k} = \frac{{(1 + \sqrt{5})}^{k} - {(1 - \sqrt{5})}^{k}}{2^{k} \sqrt{5}} f_{3} = \frac{{(1 + \sqrt{5})}^{3} - {(1 - \sqrt{5})}^{3}}{2^{3} \sqrt{5}} f_{3} = \frac{16 \sqrt{5}}{8 \sqrt{5}} f_{3} = 2$

So the third term of the sequence is $f_{3} = 2 .$

Step 5. The fourth term of the sequence.

Substitute $k = 4$ in the sequence for the fourth term.

$f_{k} = \frac{{(1 + \sqrt{5})}^{k} - {(1 - \sqrt{5})}^{k}}{2^{k} \sqrt{5}} f_{4} = \frac{{(1 + \sqrt{5})}^{4} - {(1 - \sqrt{5})}^{4}}{2^{4} \sqrt{5}} f_{4} = \frac{48 \sqrt{5}}{16 \sqrt{5}} f_{4} = 3$

So the fourth term of the sequence is $f_{4} = 3 .$

Step 6. The fifth term of the sequence.

Substitute $k = 5$ in the sequence for the fifth term.

$f_{k} = \frac{{(1 + \sqrt{5})}^{k} - {(1 - \sqrt{5})}^{k}}{2^{k} \sqrt{5}} f_{5} = \frac{{(1 + \sqrt{5})}^{5} - {(1 - \sqrt{5})}^{5}}{2^{5} \sqrt{5}} f_{5} = \frac{160 \sqrt{5}}{32 \sqrt{5}} f_{5} = 5$

So the fifth term of the sequence is $f_{5} = 5 .$

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The Fibonacci numbers may be computed with the formula
$f_{k} = \frac{{(1 + \sqrt{5})}^{k} - {(1 - \sqrt{5})}^{k}}{2^{k} \sqrt{5}}$
Use this formula to compute f $f_{1}, f_{2}, f_{3}, f_{4}, and f_{5} .$ (Imagine computing $f_{100}$ .)

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The first term of the sequence.

Step 3. The second term of the sequence.

Step 4. The third term of the sequence.

Step 5. The fourth term of the sequence.

Step 6. The fifth term of the sequence.

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The Fibonacci numbers may be computed with the formulafk=1+5k-1-5k2k5Use this formula to compute ff1,f2,f3,f4,andf5.(Imagine computing f100.)

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The first term of the sequence.

Step 3. The second term of the sequence.

Step 4. The third term of the sequence.

Step 5. The fourth term of the sequence.

Step 6. The fifth term of the sequence.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Probability and Statistics

Calculus

Decision Maths

Applied Mathematics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

The Fibonacci numbers may be computed with the formula
$f_{k} = \frac{{(1 + \sqrt{5})}^{k} - {(1 - \sqrt{5})}^{k}}{2^{k} \sqrt{5}}$
Use this formula to compute f $f_{1}, f_{2}, f_{3}, f_{4}, and f_{5} .$ (Imagine computing $f_{100}$ .)