Round to the ten millions place of the number 329,530,682

Find the nth term of the sequence 7,25,51,85,127

olya-2409 [2.1K]

Let a (n) denote the n-th term of the given sequence.

Check the forward differences, and denote the n-th difference by b (n). That is,

b (n) = a (n + 1) - a (n)

These so-called first differences are

b (1) = a (2) - a (1) = 25 - 7 = 18

b (2) = a (3) - a (2) = 51 - 25 = 26

b (3) = a (4) - a (3) = 85 - 51 = 34

b (4) = a (5) - a (4) = 127 - 85 = 42

Now consider this sequence of differences,

18, 26, 34, 42, …

and notice that the difference between consecutive terms in this sequence b is 8:

26 - 18 = 8

34 - 26 = 8

42 - 34 = 8

and so on. This means b is an arithmetic sequence, and in particular follows the rule

b (n) = 18 + 8 (n - 1) = 8n + 10

for n ≥ 1.

So we have

a (n + 1) - a (n) = 8n + 10

or, replacing n + 1 with n,

a (n) = a (n - 1) + 8 (n - 1) + 10

a (n) = a (n - 1) + 8n + 2

We can solve for a (n) by iteratively substituting:

a (n) = [a (n - 2) + 8 (n - 1) + 2] + 8n + 2

a (n) = a (n - 2) + 8 (n + (n - 1)) + 2×2

a (n) = [a (n - 3) + 8 (n - 2) + 2] + 8 (n + (n - 1)) + 2×2

a (n) = a (n - 3) + 8 (n + (n - 1) + (n - 2)) + 3×2

and so on. The pattern should be clear; we end up with

a (n) = a (1) + 8 (n + (n - 1) + … + 3 + 2) + (n - 1)×2

The middle group is the sum,

$\displaystyle 8\sum_{k=2}^nk=8\sum_{k=1}^nk-8=\frac{8n(n+1)}2-8=4n^2+4n-8$

so that

a (n) = a (1) + (4n ² + 4n - 8) + 2 (n - 1)

a (n) = 4n ² + 6n - 3

4 0

3 years ago

A business has $10,000 in its bank account. In order to build an addition onto its office, it takes out a loan for $30,000 and d

ser-zykov [4K]

Answer:

The new account balance is $40000

Step-by-step explanation:

The number line is not given. However, the question can be solved without it.

Given

$Balance = \$10000$

$Loan = \$30000$

Required

Determine the new balance

$New\ Balance = Initial\ Balance + Loan\ Amount$

$New\ Balance = \$10000 + \$30000$

$New\ Balance = \$40000$

Hence:

The new account balance is $40000

7 0

3 years ago

The Burmese pythons can be anywhere from 16 to 23 feet long. What is the difference in length inches between the longest and sho

alina1380 [7]

16*12=192
23*12=276
276-192
=84 inches

8 0

4 years ago

Read 2 more answers

Find the rule and the graph of the function whose graph can be obtained by performing the translation 3 units left and 2 units d

Leya [2.2K]

Answer: Option a.

Step-by-step explanation:

1. You have the following parent function given in the problem above:

f(x)=x³ (This is the simplest form. We need to translate it 3 units left and 2 units down)

2. If you take the parent function and make y=f(x+3), then you have:

$y=f(x+3)=(x+3)^{3}$ (The function is shifted 3 units left on the x-axis).

3. Then you if you make y=f(x+3)-2, as following, you obtain:

$y=f(x+3)-2=(x+3)^{3}-2$ (The function is shifted 2 units down on the y-axis).

4. Therefore, that is how you obtain the final function.

The answer is the graph shown in the option a.

6 0

3 years ago

Hunter is splitting a quart of ice cream with 7 members of his family. If the quarts is split evenly, how many cups will each fa

klemol [59]

1/7

This is because there are 7 members and each person gets 1/4

7 0

3 years ago