Size of angle x in 51

Find the general term of sequence defined by these conditions.

disa [49]

Answer:

$\displaystyle a_{n} = (2)^{2n -1} - (3) ^{n-1 }$

Step-by-step explanation:

we want to figure out the general term of the following recurrence relation

$\displaystyle \rm a_{n + 2} - 7a_{n + 1} + 12a_n = 0 \: \: where : \: \:a_1 = 1 \: ,a_2 = 5,$

we are given a linear homogeneous recurrence relation which degree is 2. In order to find the general term ,we need to make it a characteristic equation i.e

${x}^{n} = c_{1} {x}^{n - 1} + c_{2} {x}^{n - 2} + c_{3} {x}^{n -3 } { \dots} + c_{k} {x}^{n - k}$

the steps for solving a linear homogeneous recurrence relation are as follows:

Create the characteristic equation by moving every term to the left-hand side, set equal to zero.
Solve the polynomial by factoring or the quadratic formula.
Determine the form for each solution: distinct roots, repeated roots, or complex roots.
Use initial conditions to find coefficients using systems of equations or matrices.

Step-1:Create the characteristic equation

${x}^{2} - 7x+ 12= 0$

Step-2:Solve the polynomial by factoring

factor the quadratic:

$( {x}^{} - 4)(x - 3) = 0$

solve for x:

$x = \rm 4 \:and \: 3$

Step-3:Determine the form for each solution

since we've two distinct roots,we'd utilize the following formula:

$\displaystyle a_{n} = c_{1} {x} _{1} ^{n } + c_{2} {x} _{2} ^{n }$

so substitute the roots we got:

$\displaystyle a_{n} = c_{1} (4)^{n } + c_{2} (3) ^{n }$

Step-4:Use initial conditions to find coefficients using systems of equations

create the system of equation:

$\begin{cases}\displaystyle 4c_{1} +3 c_{2} = 1 \\ 16c_{1} + 9c_{2} = 5\end{cases}$

solve the system of equation which yields:

$\displaystyle c_{1} = \frac{1}{2} \\ c_{2} = - \frac{1}{3}$

finally substitute:

$\displaystyle a_{n} = \frac{1}{2} (4)^{n } - \frac{1}{3} (3) ^{n }$

$\displaystyle \boxed{ a_{n} = (2)^{2n-1 } - (3) ^{n -1}}$

and we're done!

7 0

3 years ago

Please help and thank you

s2008m [1.1K]

Answer:

Answer is(-1,-1)

Step-by-step explanation:

$given \: points \: are \:( 1, - 6) \: and \: ( - 3,4) \\ midpoint \: = (\frac{x1 + x2}{2} , \frac{y1 + y2}{2} ) \\ =( \frac{1 - 3}{2} , \frac{ - 6 + 4}{2} ) \\ (\frac{ - 2}{2} , \frac{ - 2}{2} ) \\ midpoint \: = ( - 1, - 1)$

 

 HAVE A NICE DAY!

THANKS FOR GIVING ME THE OPPORTUNITY TO ANSWER YOUR QUESTION. 

3 0

3 years ago

Greg drives 26 miles on Monday and90 miles on tuesday write and expression to match the words

Georgia [21]

26+90=116miles altogether

4 0

3 years ago

Tell whether each number is rational or irrational: Pi, -1/2, the square root of 2/3, and 0.57575757

DedPeter [7]

Pi: Irrational
√(2/3): Irrational

0.57575757 is rational
and -1/2 is rational.
I hope this helps.

5 0

3 years ago

A water pitcher holds 3.5 quarts of water. How many liters of water does it hold? (1 liter = 1.06 quarts) A) 2.2 liters B) 3.3 l

Leona [35]

The answer is B. 3.3 Liters.

3.5 Quarts is 3.31224 liters.

6 0

3 years ago