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Alex17521 [72]
3 years ago
7

A.Find a formula for

Mathematics
1 answer:
Vlada [557]3 years ago
5 0

a. Notice that

1/(1*2) = 1/2 = 1 - 1/2

1/(2*3) = 1/6 = 1/2 - 1/3

1/(3*4) = 1/12 = 1/3 - 1/4

and so on, which suggests the n-th term of the sum can be written as

\dfrac1{n(n+1)}=\dfrac1n-\dfrac1{n+1}

Then the sum itself is telescoping:

\dfrac1{1\cdot2}+\dfrac1{2\cdot3}+\dfrac1{3\cdot4}+\cdots+\dfrac1{n(n+1)}

=\left(1-\dfrac12\right)+\left(\dfrac12-\dfrac13\right)+\left(\dfrac13-\dfrac14\right)+\cdots+\left(\dfrac1n-\dfrac1{n+1}\right)

=1-\dfrac1{n+1}

b. The proof is trivial:

\dfrac1n-\dfrac1{n+1}=\dfrac{n+1}{n(n+1)}-\dfrac n{n(n+1)}=\dfrac{n+1-n}{n(n+1)}=\dfrac n{n+1}

so the formula found in (a) is correct.

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Answer:

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A patient drinks 5 12ounce cups of coffee daily. The doctor recommends cutting back 60%, how many ounces of coffee can the patie
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The patient is recommended to take 24 ounces of coffee per day.

<u>Step-by-step explanation:</u>

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