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3 years ago

1²+3²+4²+.......+(2n-2)²= ⅓n(2n-1)(2n+1)

1 answer:

8 0

1² + 3² + 4² + 4(n - 1)² = ¹/₃n(2n - 1)(2n + 1)
1² + 3² + 4² + (2n - 2)² = ¹/₃n(2n - 1)(2n + 1)
  1 + 9 + 16 + (2n - 2)(2n - 2) = ¹/₃n(2n(2n + 1) - 1(2n + 1))
10 + 16 + (2n(2n - 2) - 2(2n - 2)) = ¹/₃n(2n(2n) + 2n(1) - 1(2n) - 1(1) 16 + (2n(2n) - 2n(2) - 2(2n) + 2(2)) = ¹/₃n(4n² + 2n - 2n - 1)
26 + (4n² - 4n - 4n + 4) = ¹/₃n(4n² - 1)
26 + (4n² - 8n + 4) = ¹/₃n(4n² - 1)
  26 + 4n² - 8n + 4 = ¹/₃n(4n²) - ¹/₃n(1)
  4n² - 8n + 4 + 26 = 1¹/₃n³ - ¹/₃n
  4n² - 8n + 30 = 1¹/₃n³ - ¹/₃n
  + ¹/₃n + ¹/₃n
   4n² - 7²/₃n + 30 = 1¹/₃n³
-1¹/₃n³ + 4n² - 7²/₃n + 30 = 0
-3(-1¹/₃n³ + 4n² - 7²/₃n + 30) = -3(0)
-3(-1¹/₃n³) - 3(4n²) - 3(-7²/₃n) - 3(30) = 0
  4n³ - 12n² + 23n - 90 = 0

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Write a rule for the nth term of the arithmetic sequence:<br><br> a11= 50, d = 7

ivolga24 [154]

Answer:

Required rule for $n^{th}$ is $a_{n}=7n-27$ .

Step-by-step explanation:

Given that,

$a_{11} = 50,\ \ d=7$

From the question: we have to write the $n^{th}$ term of Arithmetic sequence.

Arithmetic Sequence or Arithmetic progression (A.P) : It is a sequence which possess that difference between of two successive sequence is always constant.

$a_{1} ,a_{2},a_{3},a_{4}.....................a_{n-1},a_{n}$

where, $a_{1}$ is the first term of A.P

$d$ is the common difference.

$a_{n}$ is the last term or general term.

The above sequence to be in A.P then their common difference should be equal.

$d=a_{2}-a_{1} =a_{3}-a_{2}=a_{4} -a_{3} ..........................a_{n}-a_{n-1}$

Now, Formula of General Term is $a_{n}=a+(n-1)d$

So, $a_{11}= a+(11-1)d\\ a_{11} = a+10d$

Substituting the value of $a_{11} = 50,\ \ d=7$ we get,

$50=a+10\times7\\50=a+70\\a=-20$

Then General term ( $a_{n}$ ) of given data is

$a_{n}=-20+(n-1)7\\a_{n}=-20+7n-7\\a_{n}=7n-27$

Therefore, Required rule for $n^{th}$ is $a_{n}=7n-27$ .

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3 years ago

What is the scale of a drawing that shows a length of 2 inches when the actual length is

Softa [21]

Answer:

1:18 to scale

Step-by-step explanation:

2 goes into 36 18 times so its scaled down by 18.

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2 years ago

A "b" value of less than 1 produces a graph with exponential decay

Andru [333]

<span>If this statement is a True or False question.

Then, TRUE. A "b" value of less than 1 produces a graph with exponential decay.

We consider this formula: y = a*b^x or f(x) =a*b^x

a is the initial value, b is the rate, x is the exponent.

If b has a value of less than 1, as the exponent increases the resulting value of y decreases.

f(x) = 20(0.90)^x

f(1) = 200(0.90)</span>¹ = 180
f(2) = 200(0.90)² = 162
f(3) = 200(0.90)³ = 145.80
f(4) = 200(0.90)⁴ = 131.22
f(5) = 200(0.90)⁵ = 118.10

Exponential decay is the decrease in a quantity. As you can see in the given example the value of f(x) decreases as the value of x increases because the value of b is less than 1.

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3 years ago

Preethi writes fifteen 1s in a row and randomly writes + and - between each pair of consecutive 1s. Find the probability that th

amid [387]

Answer:

The probability that the expression Preeti wrote has the value 7 cannot be easily determined.

Step-by-step explanation:

Given that Preeti writes

111111111111111,

then randomly writes + and - between each pair of consecutive 1's.

It is not specified that the + and - signs are written to be alternating, they are said to be just 'random'. So, there are a lot of possibilities. Things like:

1+1-1+1+1-1+1-1-1-1+1+1-1+1-1 = 1

1-1-1+1+1-1-1+1+1-1+1-1+1+1-1 = -1

1-1+1-1-1+1-1+1+1-1-1+1-1+1+1 = 0

And so on. There is a lot of possibilities.

So, clearly, the probability is difficult to determine, as the expression Preeti wrote could have had the value 7 in many different forms.

Because:

1+1+1+1+1+1+1-1+1-1+1-1+1-1+1 = 7

1+1+1+1-1-1-1-1-1+1-1-1-1+1+1 = 7

And many more.

Therefore, the probability that the expression Preeti wrote has the value 7 cannot be easily determined.

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2 years ago

Rods car takes 14 gallon to fill up and gets 35 miles per gallon rod drives 17.5 miles a day can rod go a whole month with out f

hammer [34]

Answer:

Step-by-step explanation:

Rod would have to fill his care up 7 times to drive a whole month, cause a month would be 225 miles and his car holds 35 miles worth of gas.

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3 years ago