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2 years ago

How many blocks fill this prism?

Mathematics

1 answer:

zaharov [31]2 years ago

3 0

Answer : 72 blocks

To find the answer, you have to find the volume of the prism. To find the volume you multiply the length, width, and height.

6 x 3 x 4 = 72

This gives you 72 blocks.

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Determine formula of the nth term 2, 6, 12 20 30,42

nalin [4]

Check the forward differences of the sequence.

If $\{a_n\} = \{2,6,12,20,30,42,\ldots\}$ , then let $\{b_n\}$ be the sequence of first-order differences of $\{a_n\}$ . That is, for n ≥ 1,

$b_n = a_{n+1} - a_n$

so that $\{b_n\} = \{4, 6, 8, 10, 12, \ldots\}$ .

Let $\{c_n\}$ be the sequence of differences of $\{b_n\}$ ,

$c_n = b_{n+1} - b_n$

and we see that this is a constant sequence, $\{c_n\} = \{2, 2, 2, 2, \ldots\}$ . In other words, $\{b_n\}$ is an arithmetic sequence with common difference between terms of 2. That is,

$2 = b_{n+1} - b_n \implies b_{n+1} = b_n + 2$

and we can solve for $b_n$ in terms of $b_1=4$ :

$b_{n+1} = b_n + 2$

$b_{n+1} = (b_{n-1}+2) + 2 = b_{n-1} + 2\times2$

$b_{n+1} = (b_{n-2}+2) + 2\times2 = b_{n-2} + 3\times2$

and so on down to

$b_{n+1} = b_1 + 2n \implies b_{n+1} = 2n + 4 \implies b_n = 2(n-1)+4 = 2(n + 1)$

We solve for $a_n$ in the same way.

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

Then

$a_{n+1} = (a_{n-1} + 2n) + 2(n+1) \\ ~~~~~~~= a_{n-1} + 2 ((n+1) + n)$

$a_{n+1} = (a_{n-2} + 2(n-1)) + 2((n+1)+n) \\ ~~~~~~~ = a_{n-2} + 2 ((n+1) + n + (n-1))$

$a_{n+1} = (a_{n-3} + 2(n-2)) + 2((n+1)+n+(n-1)) \\ ~~~~~~~= a_{n-3} + 2 ((n+1) + n + (n-1) + (n-2))$

and so on down to

$a_{n+1} = a_1 + 2 \displaystyle \sum_{k=2}^{n+1} k = 2 + 2 \times \frac{n(n+3)}2$

$\implies a_{n+1} = n^2 + 3n + 2 \implies \boxed{a_n = n^2 + n}$

6 0

1 year ago

What is interest earned on 3500 invested at 6% compounded quarterly or 12 years

enyata [817]

$\bf \qquad \textit{Compound Interest Earned Amount}
\\\\
A=P\left(1+\frac{r}{n}\right)^{nt}
\quad 
\begin{cases}
A=\textit{accumulated amount}\\
P=\textit{original amount deposited}\to &\$3500\\
r=rate\to 6\%\to \frac{6}{100}\to &0.06\\
n=
\begin{array}{llll}
\textit{times it compounds per year}\\
\textit{quarterly, thus four}
\end{array}\to &4\\
t=years\to &12
\end{cases}
\\\\\\
A=3500\left(1+\frac{0.06}{4}\right)^{4\cdot 12}\implies A=3500(1.015)^{48}$

so, the interest earned then will be A - 3500.

6 0

2 years ago

The altitude drawn to the hypotenuse of a right triangle divides the hypotenuse into segments of lengths 4 and 12. Find the leng

Ostrovityanka [42]

Answer:

8 units.

Step-by-step explanation:

The smaller of the right triangles formed is similar to the whole triangle so

4/x = x/16 where x = the shorter leg

x^2 = 64

x = 8.

7 0

2 years ago

Pi x 5 what do you do with this I do not understand

qaws [65]

You would be multiplying pi (3.14) by the number 5

5 0

3 years ago

How is an equation like an expression? How is it different?

morpeh [17]

Answer:

See explanation.

Step-by-step explanation:

An expression is a set of terms that are combined using addition, subtraction, multiplication, or division. The following are examples of expressions:

$2x-1$