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3 years ago

Geometry #1 please will give brainliest

Mathematics

1 answer:

pentagon [3]3 years ago

5 0

Step-by-step explanation:

If two lines, segments, or rays intersect at right angles, then they are perpendicular. If two angles are complementary, then their measures add up to 90 degrees. If two angles' measures add up to 90 degrees, then they are complementary. If two angles are supplementary, then their measures add up to 180 degrees.

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C=1/21.22.23+1/22.23.24+................+1/200.201.202 . = là dấu nhân

Aneli [31]

It looks like you have to find the value of the sum,

$C = \displaystyle \frac1{21\times22\times23} + \frac1{22\times23\times24} + \cdots + \frac1{200\times201\times202}$

so that the n-th term in the sum is

$\dfrac1{(21+(n-1))\times(21+n)\times(21+(n+1))} = \dfrac1{(n+20)(n+21)(n+22)}$

for 1 ≤ n ≤ 180.

We can then write the sum as

$\displaystyle C = \sum_{n=1}^{180} \frac1{(n+20)(n+21)(n+22)}$

Break up the summand into partial fractions:

$\dfrac1{(n+20)(n+21)(n+22)} = \dfrac a{n+20} + \dfrac b{n+21} + \dfrac c{n+22}$

Combine the fractions into one with a common denominator and set the numerators equal to one another:

$1 = a(n+21)(n+22) + b(n+20)(n+22) + c(n+20)(n+21)$

Expand the right side and collect terms with the same power of n :

$1 = a(n^2+43n+462)+b(n^2+42n+440) + c(n^2+41n + 420) \\\\ 1 = (a+b+c)n^2 + (43a+42b+41c)n + 462a+440b+420c$

Then

a + b + c = 0

43a + 42b + 41c = 0

462a + 440b + 420c = 1

==> a = 1/2, b = -1, c = 1/2

Now our sum is

$\displaystyle C = \sum_{n=1}^{180} \left(\frac1{2(n+20)}-\frac1{n+21}+\frac1{2(n+22)}\right)$

which is a telescoping sum. If we write out the first and last few terms, we have

C = 1/(2×21) - 1/22 + 1/(2×23)

… … + 1/(2×22) - 1/23 + 1/(2×24)

… … + 1/(2×23) - 1/24 + 1/(2×25)

… … + 1/(2×24) - 1/25 + 1/(2×26)

… … + … - … + …

… … + 1/(2×198) - 1/199 + 1/(2×200)

… … + 1/(2×199) - 1/200 + 1/(2×201)

… … + 1/(2×200) - 1/201 + 1/(2×202)

Notice the diagonal pattern of underlined and bolded terms that add up to zero (e.g. 1/(2×23) - 1/23 + 1/(2×23) = 1/23 - 1/23 = 0). So, like a telescope, the sum collapses down to a simple sum of just six terms,

C = 1/(2×21) - 1/22 + 1/(2×22) + 1/(2×201) - 1/201 + 1/(2×202)

which we simplify further to

C = 1/42 - 1/44 - 1/402 + 1/404

C = 1,115/1,042,118 ≈ 0.00106994

4 0

3 years ago

William bought 2.35 pounds of apples. Jill bought 0.764 pounds of apples. Frank bought 2.5 times the total amount that William a

masha68 [24]

Answer:

7.785

Step-by-step explanation:

William and Jill together is 3.114 so just times that by 2.5 and your good