All categories

2 years ago

Use the binomial theorem to show that 7^n + 2 is divisible by 3, where n is a positive integer.

Mathematics

1 answer:

masha68 [24]2 years ago

7 0

By the binomial theorem,

$7^n=(6+1)^n=6^n+n6^{n-1}+\dbinom n26^{n-2}+\cdots+\dbinom n{n-2}6^2+n6^1+1$

Each term containing a factor 6 is divisible by 3, since 3 divides 6. The remaining 1 and 2 sum to 3, which is also divisible by 3.

You might be interested in

Find the midpoint of the segment between the points (3,17) and (−14,−8)

mixas84 [53]

Answer:

$\large\boxed{\left(-\dfrac{11}{2},\ \dfrac{9}{2}\right)}$

Step-by-step explanation:

The formula of a midpoint between two points A(x₁, y₁) and B(x₂, y₂):

$M_{AB}\left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)$

We have the points (3, 17) and (-14, -8).

Substitute:

$M(x,\ y)\\\\x=\dfrac{3+(-14)}{2}=\dfrac{-11}{2}\\\\y=\dfrac{17+(-8)}{2}=\dfrac{9}{2}$

5 0

2 years ago

Read 2 more answers

A sinusoidal function whose period is π/2 , maximum value is 10, and minimum value is −4 has a y-intercept of 3. What is the equ

Eddi Din [679]

Answer:

$f(\theta)=3+7sin(4\theta)$

Step-by-step explanation:

The sinusoidal function has a general form of:

$f(\theta)=Y+Asin(k\theta)$

The period is given by:

$P=\frac{2\pi}{k}\\\frac{\pi}{2}= \frac{2\pi}{k}\\k=4$

Y is the y-intercept, thus y = 3.

At the maximum value:

$f(max) = Y+A\\10 = 3+A\\A=7$

At the minimum value:

$f(min) = Y-A\\-4 = 3-A\\A=7$

Therefore, the equation of the function described is:

$f(\theta)=3+7sin(4\theta)$

6 0

2 years ago

The Great Lakes differ in both their areas (measured in square miles) and their depths. However these two dimensions do not keep

Masteriza [31]

Answer:

The order of Great Lakes according to depth is (descending order): 1. Lake Superior 2. Lake Michigan 3. Lake Ontario 4. Lake Huron 5. Lake Erie

Step-by-step explanation:

Lake Superior is by far the largest and deepest of the great Lakes. Lake Michigan is exceeded in depth only by Lake Superior, but it is exceeded in area by both Lakes Superior and Huron. Lake Ontario, which is the smallest in area, is deeper than both Lakes Huron and Erie. Lake Erie is larger than Lake Ontario but it is not only shallower than Huron; it is also shallower than Ontario. So, the order of Great Lakes according to depth is (descending order): 1. Lake Superior 2. Lake Michigan 3. Lake Ontario 4. Lake Huron 5. Lake Erie

3 0

2 years ago

Bilal's fishing line was on a spool with a radius of 4 \text{ cm}4 cm4, start text, space, c, m, end text. Suddenly, a fish pull

Fantom [35]

Answer:

The distance the fish pulled the fishing line is <u>401.92 cm.</u>

Step-by-step explanation:

Given:

Radius of fishing spool = 4 cm.

Fish pulled on the line, and the spool spun 16 times before Bilal began to reel in the fish.

Now, to find the distance the fish pulled the fishing line.

So, to get the circumference of the spool first we put formula: