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vlabodo [156]
3 years ago
7

Prove that for any natural n the value of the expression (n+7)^2–n^2 is divisible by 7.

Mathematics
1 answer:
Afina-wow [57]3 years ago
3 0

Step-by-step explanation:

(n + 7)² – n²

n² + 14n + 49 – n²

14n + 49

7 (2n + 7)

7 is a factor, so for any natural n, (n + 7)² – n² is a multiple of 7.

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Consider the sequence given by a plus 8 pattern: 2, 10, 18, 26, ….
Finger [1]

Answer:

n = 1 second formula

n = 0 first formula

Step-by-step explanation:

I answer this in the other question you put, here it is again.

This is easy to get. We know the sequence cause it follows a pattern of 8, so let's try some values of n from 1 to 4, to get those numbers with the first formula:

n = 1,2,3,4

f(1) = 8(1) + 2 = 10

f(2) = 8(2) + 2 = 18

f(3) = 8(3) + 2 = 26

f(4) = 8(4) + 2 = 34

As you can see, with the first formula, the first term is 10, and not 2. The only way to get 2 with n = 1 is with the second formula:

f(1) = 8(1) - 6 = 2

f(2) = 8(2) - 6 = 10

f(3) = 8(3) - 6 = 18

f(4) = 8(4) - 6 = 26

With n = 1, the second formula was better and correct.

The first formula could be right only beggining with n = 0. Here is the proof:

f(0) = 8(0) + 2 = 2

6 0
3 years ago
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