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Chapter 7: Q. 10 (page 591)

The Lucas numbers are defined recursively as follows:
$L_{1} = 1, L_{2} = 3,$ and $L_{k} = L_{k - 2} + L_{k - 1}$ for $k 鈮� 3$ .
What are $L_{3}, L_{4}, L_{5},$ and $L_{6}$ ?

Short Answer

Expert verified

The required values are $L_{3} = 4, L_{4} = 7, L_{5} = 11, L_{6} = 18$ .

Step by step solution

01

Step 1. Given information

$L_{1} = 1, L_{2} = 3$ .

L_{k} = L_{k - 2} + L_{k - 1}

where

k 鈮� 3

.

02

Step 2. Substitute k=3 and L1=1,L2=3 in the given sequence.

$L_{3} = L_{3 - 2} + L_{3 - 1} = L_{1} + L_{2} = 1 + 3 = 4$

Substitute $k = 4$ and $L_{2} = 3, L_{3} = 4$ in the given sequence.

$L_{4} = L_{4 - 2} + L_{4 - 1} = L_{2} + L_{3} = 3 + 4 = 7$

03

Step 3. Substitute k=5 and L3=4,L4=7 in the given sequence.

$L_{5} = L_{5 - 2} + L_{5 - 1} = L_{3} + L_{4} = 4 + 7 = 11$

Substitute $k = 6$ and $L_{4} = 7, L_{5} = 11$ in the given sequence.

$L_{6} = L_{6 - 2} + L_{6 - 1} = L_{4} + L_{5} = 7 + 11 = 18$

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Most popular questions from this chapter

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

${鈭�}_{k = 2}^{鈭�} 鈥� \frac{1}{k \sqrt{\ln 鈦� k}}$

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

${鈭�}_{k = 1}^{鈭�} \frac{k}{3 k^{2} + 100}$

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

${鈭�}_{k = 3}^{鈭�} 鈥� \frac{1}{(k 鈭� 2)^{2}}$

Given that $a_{0} = - 3, a_{1} = 5, a_{2} = - 4, a_{3} = 2$ and ${鈭� a_{k}}_{k = 2}^{鈭�} = 7$ , find the value ofrole="math" localid="1648828282417" ${鈭� a_{k}}_{k = 1}^{鈭�}$ .

See all solutions

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