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daser333 [38]
2 years ago
7

Use mathematical induction to prove that the statement is true for every positive integer n.

Mathematics
1 answer:
Mice21 [21]2 years ago
7 0

Induction on n

Base case: n = 0

the left side is 0, and the right side is 0, so the equality holds.

Now, suppose that is true for n, and you need to show that the equality holds for n+1

So, write the left side

8 + 16 +24 + ... + 8n + 8(n+1) (*)

By hypothesis, 8+16+24+ ...+ 8n = 4n(n+1)

so, in (*) we have

4n(n+1) +8(n+1) = (4n+8)(n+1) = 4(n+2)(n+1) = 4(n+1)(n+2). So, this is true for every natural number n.


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Kelly price sells college textbooks on commission. She gets 8 % on the first $7500 of​ sales, 16% on the next $7500 of​ sales, a
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Answer:

The commission would be $ 920.

Step-by-step explanation:

Given,

Sales in 1997 ​= $ 9500,

Since, there is a commision of 8 % on the first $ 7500 of​ sales, 16% on the next $ 7500 of​ sales, and 20% on sales over $ 15,000,

If x represents the total sales,

Commission for x ≤ 7500 = 8% of x = 0.08x,

Commission for 7500 < x ≤ 15000 = 8% of 7500 + 16% of (x-7500)

= 600 + 0.16(x-7500),

Commission for 7500 < x ≤ 15000 = 8% of 7500 + 16% of 7500 + 20% of (x-15000)

= 600 + 1200 + 0.20(x-15000)

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Thus, the function that shows the given situation,

f(x)=\left\{\begin{matrix}0.08x & x \leq 7500\\ 600+0.16(x-7500) & 7500 < x \leq 15000\\ 1800+0.20(x-15000) & 15000 < x\end{matrix}\right.

Since, 9500 lies between 7500 and 15000,

Therefore, the commission would be,

f(9500)= 600 + 0.16(9500 - 7500) = 600 + 320 = $ 920

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