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2 years ago

What is Integration ?

Mathematics

2 answers:

jek_recluse [69]2 years ago

7 0

Answer:

Integration is the act of bringing together smaller components into a single system that functions as one. These links usually are established between the components of the process and control layer of each system to promote the free flow of data across systems.

Veseljchak [2.6K]2 years ago

3 0

Answer:

Integration is the action or process of integrating.

Step-by-step explanation:

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Find the sum of the first 25 terms in this geometric series: 8 + 6 + 4.5...

Ksivusya [100]

Step-by-step explanation:

Given the geometric sequence

8 + 6 + 4.5...

A geometric sequence has a constant ratio and is defined by

$a_n=a_1\cdot r^{n-1}$

$\mathrm{Compute\:the\:ratios\:of\:all\:the\:adjacent\:terms}:\quad \:r=\frac{a_{n+1}}{a_n}$

$\frac{6}{8}=\frac{3}{4},\:\quad \frac{4.5}{6}=\frac{3}{4}$

$\mathrm{The\:ratio\:of\:all\:the\:adjacent\:terms\:is\:the\:same\:and\:equal\:to}$

$r=\frac{3}{4}$

$\mathrm{The\:first\:element\:of\:the\:sequence\:is}$

$a_1=8$

$\mathrm{Therefore,\:the\:}n\mathrm{th\:term\:is\:computed\:by}\:$

$a_n=8\left(\frac{3}{4}\right)^{n-1}$

$\mathrm{Geometric\:sequence\:sum\:formula:}$

$a_1\frac{1-r^n}{1-r}$

$\mathrm{Plug\:in\:the\:values:}$

$n=25,\:\spacea_1=8,\:\spacer=\frac{3}{4}$

$=8\cdot \frac{1-\left(\frac{3}{4}\right)^{25}}{1-\frac{3}{4}}$

$\mathrm{Multiply\:fractions}:\quad \:a\cdot \frac{b}{c}=\frac{a\:\cdot \:b}{c}$

$=\frac{\left(1-\left(\frac{3}{4}\right)^{25}\right)\cdot \:8}{1-\frac{3}{4}}$

$=\frac{8\left(-\left(\frac{3}{4}\right)^{25}+1\right)}{\frac{1}{4}}$

$\mathrm{Apply\:exponent\:rule}:\quad \left(\frac{a}{b}\right)^c=\frac{a^c}{b^c}$

$=\frac{8\left(-\frac{3^{25}}{4^{25}}+1\right)}{\frac{1}{4}}$

$\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{a}{\frac{b}{c}}=\frac{a\cdot \:c}{b}$

$=\frac{\left(1-\frac{3^{25}}{4^{25}}\right)\cdot \:8\cdot \:4}{1}$

$\mathrm{Multiply\:the\:numbers:}\:8\cdot \:4=32$

$=\frac{32\left(-\frac{3^{25}}{4^{25}}+1\right)}{1}$

$=\frac{32\cdot \frac{4^{25}-3^{25}}{4^{25}}}{1}$ ∵ $\mathrm{Join}\:1-\frac{3^{25}}{4^{25}}:\quad \frac{4^{25}-3^{25}}{4^{25}}$

$=32\cdot \frac{4^{25}-3^{25}}{4^{25}}$

$=\frac{\left(4^{25}-3^{25}\right)\cdot \:32}{4^{25}}$

$=\frac{2^5\left(4^{25}-3^{25}\right)}{2^{50}}$ ∵ $\mathrm{Factor}\:32:\ 2^5$ , $\mathrm{Factor}\:4^{25}:\ 2^{50}$

$=\frac{4^{25}-3^{25}}{2^{45}}$ ∵ $\mathrm{Cancel\:}\frac{\left(4^{25}-3^{25}\right)\cdot \:2^5}{2^{50}}:\quad \frac{4^{25}-3^{25}}{2^{45}}$

$\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{a\pm \:b}{c}=\frac{a}{c}\pm \frac{b}{c}$

$=\frac{4^{25}}{2^{45}}-\frac{3^{25}}{2^{45}}$

$=32-\frac{3^{25}}{2^{45}}$ ∵ $\frac{4^{25}}{2^{45}}=32$

$=32-0.024$ ∵ $\frac{3^{25}}{2^{45}}=0.024$

$=31.98$

Therefore, the sum of the first 25 terms in this geometric series: 31.98

3 0

3 years ago

81(p+q)^2-9p-9q Factor this

polet [3.4K]

Answer:

9(p + q)(9p + 9q - 1)

Step-by-step explanation:

Given

81(p + q)² - 9p - 9q ← factor out - 9 from these 2 terms

= 81(p + q)² - 9(p + q) ← factor out 9(p + q) from each term

= 9(p + q)(9(p + q) - 1)

= 9(p + q)(9p + 9q - 1)

8 0

3 years ago

Read 2 more answers

Each basket can hold 2.5 pounds of apples . how many pounds can 7 baskets hold?

schepotkina [342]

You must multiply 2.5 by 7 to get the answer which is 17.5 pounds

8 0

3 years ago

Read 2 more answers

Find an equation in point-slope form for the line having the slope m= -5 and containing the point (7,2).

Juli2301 [7.4K]

Answer:

y - 2 = - 5(x - 7)

Step-by-step explanation:

The equation of a line in point- slope form is

y - b = m(x - a)

where m is the slope and (a, b) a point on the line

Here m = - 5 and (a, b) = (7, 2) , thus

y - 2 = - 5(x - 7) ← equation in point- slope form

5 0

3 years ago

Read 2 more answers

Solve the equation. 33 = p – 6.71 A. –39.71 B. –26.29 C. 39.71 D. 26.29

jeka57 [31]

Answer

C. 39.71

Explanation

33 = p - 6.71

The first step is to make the like terms to be on the same side.

Add 6.71 on both sides of the eqution

33 + 6.71 = p - 6.71 + 6.71

39.71 = p

∴ p = 39.71

4 0

3 years ago