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hoa [83]
2 years ago
14

Use induction to prove: For every integer n > 1, the number n5 - n is a multiple of 5.

Mathematics
1 answer:
nignag [31]2 years ago
3 0

Answer:

we need to prove : for every integer n>1, the number n^{5}-n is a multiple of 5.

1) check divisibility for n=1, f(1)=(1)^{5}-1=0  (divisible)

2) Assume that f(k) is divisible by 5, f(k)=(k)^{5}-k

3) Induction,

f(k+1)=(k+1)^{5}-(k+1)

=(k^{5}+5k^{4}+10k^{3}+10k^{2}+5k+1)-k-1

=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k

Now, f(k+1)-f(k)

f(k+1)-f(k)=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k-(k^{5}-k)

f(k+1)-f(k)=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k-k^{5}+k

f(k+1)-f(k)=5k^{4}+10k^{3}+10k^{2}+5k

Take out the common factor,

f(k+1)-f(k)=5(k^{4}+2k^{3}+2k^{2}+k)      (divisible by 5)

add both the sides by f(k)

f(k+1)=f(k)+5(k^{4}+2k^{3}+2k^{2}+k)

We have proved that difference between f(k+1) and f(k) is divisible by 5.

so, our assumption in step 2 is correct.

Since f(k) is divisible by 5, then f(k+1) must be divisible by 5 since we are taking the sum of 2 terms that are divisible by 5.

Therefore, for every integer n>1, the number n^{5}-n is a multiple of 5.

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Answer:

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We start at (2,0).

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2 years ago
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Kruka [31]

Answer:

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Step-by-step explanation:

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3 years ago
Which of the algebraic expressions are linear expressions? Select two answers.
cupoosta [38]

Answer:

Option A and Option E

Step-by-step explanation:

By definition linear expressions  have the following form:

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To answer the question, identify among the options the expressions that have this form:

Option A.

-9y

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3 years ago
Solve and check:8a-4=25+6(a-8)
OlgaM077 [116]
8a - 4 = 25 + 6 (a - 8)

(distribute the 6)

8a - 4 = 25 + 6a - 48

(combine like terms 25 - 48)

8a - 4 = 6a - 23

( subtract 6a from both sides)

2a - 4 = -23

(add 4 to both sides)

2a = -19

(divide both sides by 2)

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////CHECK/////

8 (-9.5) - 4 = 25 + 6 ( -9.5 -8)

(plug each side into a calculator)

-80 = -80

FINAL ANSWER:

a = -9.5
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3 years ago
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gizmo_the_mogwai [7]

Answer:

£265.10

Step-by-step explanation:

200+33.60+31.50=£265.10

  • <em>42×</em><u>8=336=10%=33.60</u>
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3 0
3 years ago
Read 2 more answers
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