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Yuri [45]
3 years ago
15

The length of a rectangle is 5 ft more than its width. If the area is 104 sq ft, find the length and width

Mathematics
1 answer:
Kay [80]3 years ago
8 0
Area = l * w  104  =(w+5) * w w2 +5w -104 =0 (w+13)(w-8)=0w=13 and w=8so width =8length=13
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Create an arithmetic sequence where term 1 is 1000
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a_{n} = 1000 + 3(n-1)

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what are the digits that repeat in the smallest sequence of repeating digits and the decimal equivalent of 1/9
aleksklad [387]

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\frac{1}{9} = 0.1111111... = 0.1 (repeating)

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Given the similarity statement<br> AJKL ANOP, what's the<br> corresponding side of ON ?
levacccp [35]

Answer:

ON = KJ

Step-by-step explanation:

JKL = NOP

We know the angles match

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Find the sequence of differences for the given sequence. Next, find a closed form solution for the following sequence. (Be sure
padilas [110]

Let a_n denote the given sequence with n\ge1:

\{a_n\}_{n\ge1}=\{1,5,11,19,29,41,\ldots\}

Let b_n be the sequence of the forward differences of a_n, so that b_n=a_{n+1}-a_n for n\ge1:

\{b_n\}_{n\ge1}=\{4,6,8,10,12,\ldots\}

b_n follows an arithmetic progression with a difference of 2 between terms, so that

b_n=4+2(n-1)=2n+2

Then we have

2n+2=a_{n+1}-a_n\implies a_{n+1}=a_n+2n+2

so that a_n is given recursively by

\begin{cases}a_1=1\\a_{n+1}=a_n+2(n+1)&\text{for }n\ge1\end{cases}

By substitution, we can try to find a pattern:

a_2=a_1+2\cdot2

a_3=a_2+2\cdot3=a_1+2(2+3)

a_4=a_3+2\cdot4=a_1+2(2+3+4)

a_5=a_4+2\cdot5=a_1+2(2+3+4+5)

and so on, with the general pattern

a_n=a_1+2(2+3+4+\cdots+n)

and since a_1=1 we can write this as

a_n=1+2(2+3+4+\cdots+n)

a_n=-1+2+2(2+3+4+\cdots+n)

a_n=2(1+2+3+4+\cdots+n)-1

Recall that

\displaystyle\sum_{i=1}^ni=1+2+3+\cdots+n=\dfrac{n(n+1)}2

Then

a_n=2\dfrac{n(n+1)}2-1\implies\boxed{a_n=n^2+n-1}

6 0
3 years ago
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