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3 years ago

A square has a side lengths of 9, use the pythagorean theorem to find the length of the diagonal

Mathematics

1 answer:

Semmy [17]3 years ago

5 0

diagonal = 9√2 ≈ 12.73

the diagonal splits the square into 2 right triangles with the hypotenuse being the diagonal of the square

using Pythagoras' identity

d = √(9² + 9²) = √162 = 9√2 ≈ 12.73 ( to 2 dec. places )

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• 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.

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

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From the given recurrence, we know

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

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

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

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

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

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

which is what we needed. QED

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