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3 years ago

Find two ratios equivalent to the ratio in Exercise 1.

Mathematics

1 answer:

pshichka [43]3 years ago

8 0

Answer:

12:26, 24:52 :D

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If S_1=1,S_2=8 and S_n=S_n-1+2S_n-2 whenever n≥2. Show that S_n=3⋅2n−1+2(−1)n for all n≥1.

Snezhnost [94]

You can try to show this by induction:

• According to the given closed form, we have $S_1=3\times2^{1-1}+2(-1)^1=3-2=1$ , which agrees with the initial value S₁ = 1.

• Assume the closed form is correct for all n up to n = k. In particular, we assume

$S_{k-1}=3\times2^{(k-1)-1}+2(-1)^{k-1}=3\times2^{k-2}+2(-1)^{k-1}$

and

$S_k=3\times2^{k-1}+2(-1)^k$

We want to then use this assumption to show the closed form is correct for n = k + 1, or

$S_{k+1}=3\times2^{(k+1)-1}+2(-1)^{k+1}=3\times2^k+2(-1)^{k+1}$

From the given recurrence, we know

$S_{k+1}=S_k+2S_{k-1}$

so that

$S_{k+1}=3\times2^{k-1}+2(-1)^k + 2\left(3\times2^{k-2}+2(-1)^{k-1}\right)$

$S_{k+1}=3\times2^{k-1}+2(-1)^k + 3\times2^{k-1}+4(-1)^{k-1}$

$S_{k+1}=2\times3\times2^{k-1}+(-1)^k\left(2+4(-1)^{-1}\right)$

$S_{k+1}=3\times2^k-2(-1)^k$

$S_{k+1}=3\times2^k+2(-1)(-1)^k$

$\boxed{S_{k+1}=3\times2^k+2(-1)^{k+1}}$

which is what we needed. QED

6 0

2 years ago

Evaluate the function Find f(-10) if f(x) = 8x – 3

erma4kov [3.2K]

Answer:

-83

Step-by-step explanation:

so plug in 10 in 8x. So it is 10*-8 -3.

-80-3=-83

8 0

2 years ago

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Four times a number, minus 9, is equal to three times the number, plus 6

marshall27 [118]

4 x p - 9 = 3 x p + 6
Hope it helped

7 0

2 years ago

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In 5 seconds an athlete runs 40 feet. what is the average speed in miles per hour?

NikAS [45]

Answer:

12 x 40 = 480 feet per minute.

480 x 60 = 28800 feet per hour.

28800/5280 = 5.45 miles per hour.

Hope that helps.Thank you!!!

6 0

2 years ago

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What is 967 divided by 8

jeka94

Answer:

the answer would be 120.875

hope this helps :)

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3 years ago

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