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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

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

valkas [14]

$a_{n} = 1000 + 3(n-1)$

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3 years ago

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

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4 years ago

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

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Step-by-step explanation:

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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 $n\ge1$ :

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

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

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

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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