CALCU-Evaluate _k=3^5 3^kAnswer:

Solution by Steps

step 1

We start by evaluating the summation

\\sum_{k=3}^{5} 3^{k}

. This means we will calculate

3^{3}

,

3^{4}

, and

3^{5}

and then sum these values

step 2

Calculate

3^{3} = 27

step 3

Calculate

3^{4} = 81

step 4

Calculate

3^{5} = 243

step 5

Now, we sum these results:

27 + 81 + 243 = 351

Answer

351

Key Concept

Summation of powers of a number

Explanation

The summation involves calculating the powers of 3 from k=3 to k=5 and then adding them together to get the final result.

Solution by Steps

step 1

The first three terms of the geometric progression (GP) are given as

-1, 3, -9

. We can identify the first term

a = -1

and the common ratio

r

. To find

r

, we use the formula

r = \frac{a_2}{a_1} = \frac{3}{-1} = -3

step 2

The general formula for the

n^{th}

term of a GP is given by

a_n = a \cdot r^{n-1}

. Here, we need to find the

13^{th}

term, so we will substitute

n = 13

,

a = -1

, and

r = -3

into the formula

step 3

Substituting the values, we have

a_{13} = -1 \cdot (-3)^{13-1} = -1 \cdot (-3)^{12}

step 4

Now, we calculate

(-3)^{12}

. Since

(-3)^{12} = 3^{12}

(as the exponent is even), we find

3^{12} = 531441

. Thus,

a_{13} = -1 \cdot 531441 = -531441

Answer

-531441

Key Concept

A geometric progression is defined by a constant ratio between consecutive terms.

Explanation

The

13^{th}

term is calculated using the formula for the

n^{th}

term of a GP, which incorporates the first term and the common ratio. The result shows how the terms grow exponentially based on the common ratio.

Solution by Steps

step 1

The first three terms of the arithmetic series are given as

-12, -8, -4

. We can identify the first term

a = -12

and the common difference

d

. To find

d

, we calculate

d = -8 - (-12) = 4

step 2

The formula for the sum of the first

n

terms of an arithmetic series is given by

S_n = \frac{n}{2} (2a + (n-1)d)

. Here, we need to find the sum of the first 45 terms, so

n = 45

step 3

Substituting the values into the formula:

S_{45} = \frac{45}{2} (2(-12) + (45-1)(4))

step 4

Simplifying the expression:

S_{45} = \frac{45}{2} (-24 + 44) = \frac{45}{2} (20) = 45 \cdot 10 = 450

Answer

450

Key Concept

The sum of an arithmetic series can be calculated using the formula for the sum of the first n terms.

Explanation

The answer is derived by identifying the first term and common difference, then applying the sum formula for arithmetic series. The final sum of the first 45 terms is 450.

Solution by Steps

step 1

We need to find the sum of all integers from 300 to 700 that are divisible by 7. First, we identify the smallest integer greater than or equal to 300 that is divisible by 7, which is 301

step 2

Next, we find the largest integer less than or equal to 700 that is divisible by 7, which is 700

step 3

The integers divisible by 7 between 301 and 700 form an arithmetic sequence where the first term

a = 301

and the last term

l = 700

. The common difference

d = 7

step 4

To find the number of terms

n

in this sequence, we use the formula for the

n

-th term of an arithmetic sequence:

l = a + (n-1)d

. Solving for

n

:

700 = 301 + (n-1) \cdot 7

gives

n = 58

step 5

The sum

S_n

of the first

n

terms of an arithmetic series is given by

S_n = \frac{n}{2} (a + l)

. Substituting the values:

S_{58} = \frac{58}{2} (301 + 700) = 29 \cdot 1001 = 29029

Answer

29029

Key Concept

Sum of an arithmetic series

Explanation

The sum of integers divisible by a number within a range can be calculated using the properties of arithmetic sequences. Here, we identified the first and last terms, calculated the number of terms, and applied the sum formula.

Solution by Steps

step 1

We know that in an arithmetic series, the

n^{th}

term can be expressed as

T_n = a + (n-1)d

, where

a

is the first term and

d

is the common difference. Given

T_3 = 59

and

T_{11} = 155

, we can set up two equations:

a + 2d = 59

and

a + 10d = 155

step 2

Subtract the first equation from the second to eliminate

a

:

(a + 10d) - (a + 2d) = 155 - 59

, which simplifies to

8d = 96

. Thus, we find

d = 12

step 3

Now, substitute

d = 12

back into the first equation:

a + 2(12) = 59

, which gives

a + 24 = 59

. Therefore,

a = 35

step 4

The formula for the sum of the first

n

terms of an arithmetic series is given by

S_n = \frac{n}{2} (2a + (n-1)d)

. We need to find

n

such that

S_n = 3255

step 5

Setting up the equation:

3255 = \frac{n}{2} (2(35) + (n-1)(12))

. This simplifies to

3255 = \frac{n}{2} (70 + 12n - 12)

, or

3255 = \frac{n}{2} (12n + 58)

step 6

Multiplying both sides by 2 to eliminate the fraction:

6510 = n(12n + 58)

. Rearranging gives

12n^2 + 58n - 6510 = 0

step 7

We can solve this quadratic equation using the quadratic formula

n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

, where

a = 12

,

b = 58

, and

c = -6510

step 8

Calculating the discriminant:

b^2 - 4ac = 58^2 - 4(12)(-6510) = 3364 + 313680 = 317044

. Thus,

n = \frac{-58 \pm \sqrt{317044}}{24}

step 9

Approximating

\sqrt{317044} \approx 563.1

, we find

n = \frac{-58 \pm 563.1}{24}

. The positive solution gives

n \approx \frac{505.1}{24} \approx 21.04

. Since

n

must be a whole number, we round to

n = 21

Answer

21

Key Concept

An arithmetic series is defined by a first term and a common difference, allowing us to find any term and the sum of the series.

Explanation

We determined the first term and common difference, then used the sum formula to find the number of terms needed to reach a specific sum.

Generated Graph

Solution by Steps

step 1

We know that the terms

5

,

(4+x)

, and

17

are part of an arithmetic sequence. In an arithmetic sequence, the difference between consecutive terms is constant. Thus, we can set up the equation:

(4+x) - 5 = 17 - (4+x)

step 2

Simplifying the equation from step 1 gives us:

(4+x) - 5 = 17 - 4 - x

. This simplifies to

x - 1 = 13 - x

step 3

Adding

x

to both sides results in:

2x - 1 = 13

. Adding

1

to both sides gives us

2x = 14

step 4

Dividing both sides by

2

yields:

x = 7

Answer

7

Key Concept

Arithmetic Sequence

Explanation

In an arithmetic sequence, the difference between consecutive terms is constant, allowing us to set up equations to find unknown terms. Here, we found

x

by establishing the relationship between the terms.

Generated Graph

Solution by Steps

step 1

We start with the formula for

S_n

:

S_n = \frac{1}{6} n(n+1)(2n+1)

. To find

T_n

, we use the definition

T_n = S_n - S_{n-1}

. Thus, we need to calculate

S_4

and

S_3

step 2

Calculate

S_4

:

S_4 = \frac{1}{6} \cdot 4 \cdot (4+1) \cdot (2 \cdot 4 + 1) = \frac{1}{6} \cdot 4 \cdot 5 \cdot 9 = \frac{180}{6} = 30

step 3

Calculate

S_3

:

S_3 = \frac{1}{6} \cdot 3 \cdot (3+1) \cdot (2 \cdot 3 + 1) = \frac{1}{6} \cdot 3 \cdot 4 \cdot 7 = \frac{84}{6} = 14

step 4

Now, we find

T_4

:

T_4 = S_4 - S_3 = 30 - 14 = 16

Answer

16

Key Concept

Understanding the relationship between the sums of sequences and their differences.

Explanation

The value of

T_4

is derived from the difference between the sums of the first four and three terms of the sequence, leading to the result of 16.

Generated Graph

Solution by Steps

step 2

The common ratio

r

of the series can be expressed as

r = \frac{(x + \frac{1}{2})}{(x + \frac{1}{2})} = 2

for the second term and

r = 2\left(x + \frac{1}{2}\right)

for subsequent terms. Thus, we need |2\left(x + \frac{1}{2}\right)| < 1

step 3

This inequality simplifies to -1 < 2\left(x + \frac{1}{2}\right) < 1 . Dividing the entire inequality by 2 gives -\frac{1}{2} < x + \frac{1}{2} < \frac{1}{2}

step 4

Subtracting

\frac{1}{2}

from all parts of the inequality results in -1 < x < 0 . Thus, the range of

x

that allows the limiting sum to exist is -1 < x < 0

[1] Answer

C

Key Concept

Convergence of Geometric Series

Explanation

A geometric series converges if the absolute value of the common ratio is less than one. This condition helps determine the valid range for the variable in the series.

Solution by Steps

step 1

The truck travels 200 meters to the first location, then returns to the base. The total distance for this trip is

200 + 200 = 400

meters

step 2

The truck then travels 150 meters to the second location, returns to the base, adding another

150 + 150 = 300

meters

step 3

Finally, the truck travels 5 kilometers (or 5000 meters) to the last location and returns to the base, adding

5000 + 5000 = 10000

meters

step 4

Now, we sum all the distances traveled:

400 + 300 + 10000 = 10600

meters

step 5

To convert meters to kilometers, we divide by 1000:

10600 \div 1000 = 10.6

kilometers

Answer

10.6

Key Concept

Understanding distance traveled in a round trip scenario.

Explanation

The total distance is calculated by summing the distances for each trip to and from the base, then converting the final result from meters to kilometers.

Solution by Steps

step 1

Leinad deposits $1000 at the beginning of each year for 25 years at an 8% interest rate compounded annually. The future value of these deposits can be calculated using the formula for the future value of a series: $ FV = P \cdot \frac{(1 + r)^n - 1}{r} $, where $ P $ is the annual deposit, $ r $ is the interest rate, and $ n $ is the number of deposits

step 2

For the first 10 years, the future value is calculated as follows:

FV_{10} = 1000 \cdot \frac{(1 + 0.08)^{10} - 1}{0.08}

. This gives us

FV_{10} = 1000 \cdot \frac{(1.08)^{10} - 1}{0.08}

. Calculating

(1.08)^{10} \approx 2.1589

, we find

FV_{10} \approx 1000 \cdot \frac{2.1589 - 1}{0.08} \approx 1000 \cdot 14.48625 \approx 14486.25

step 3

At the beginning of the 11th year, Leinad transfers this amount to another bank with a 9% interest rate. The future value of this amount after 15 more years is calculated using the formula

FV = PV \cdot (1 + r)^n

, where

PV

is the present value. Thus,

FV_{15} = 14486.25 \cdot (1 + 0.09)^{15}

. Calculating

(1.09)^{15} \approx 4.367

, we find

FV_{15} \approx 14486.25 \cdot 4.367 \approx 63200.56

step 4

Rounding to the nearest dollar, the total amount Leinad will have at the end of 25 years is approximately

63201

Answer

63201

Key Concept

Future value of a series with compound interest

Explanation

The answer is derived from calculating the future value of annual deposits compounded at different interest rates over specified periods. The total amount reflects the growth of the initial deposits over time.

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