7.03

1. write the sum using summation notation, assuming the suggested pattern continues.
1 - 3 + 9 - 27 +

a) summation of one times three to the power of n from n equals zero to infinity
b) summation of one times three to the power of the quantity n plus one from n equals zero to infinity
c) summation of one times negative three to the power of n from n equals zero to infinity
d) summation of one times negative three to the power of the quantity n plus one from n equals zero to infinity

2. write the sum using summation notation, assuming the suggested pattern continues.
-4 + 5 + 14 + 23 + + 131

a) summation of the quantity negative four plus nine n from n equals zero to fifteen
b) summation of negative thirty six times n from n equals zero to fifteen
c) summation of the quantity negative four plus nine n from n equals zero to infinity
d) summation of negative thirty six times n from n equals zero to infinity

3. write the sum using summation notation, assuming the suggested pattern continues.
25 + 36 + 49 + 64 + + n2 +

a) summation of n squared from n equals five to infinity
b) summation of n minus one squared from n equals five to infinity
c) summation of n squared from n equals six to infinity
d) summation of n plus one squared from n equals five to infinity

4. find the sum of the arithmetic sequence.
-1, 2, 5, 8, 11, 14, 17

a) 56
b) 63
c) -7
d) 20

5. find the sum of the geometric sequence.
1, one divided by four, one divided by sixteen, one divided by sixty four, one divided by two hundred and fifty six

a) 341
b) one divided by one hundred and ninety two
c) one divided by seven hundred and sixty eight
d) three hundred and fourty one divided by two hundred and fifty six

6. find the sum of the first 8 terms of the sequence. show all work for full credit.
1, -3, -7, -11,

This are 6 question and 6 answers

Problem 1. Write the sum using summation notation, assuming the suggested pattern continues.
1 - 3 + 9 - 27 + ...

option C) summation of one times negative three to the power of n from n equals zero to infinity

Explanation:

1) Sequence: 1 - 3 + 9 - 27: given

2) ratio of two consecutive terms:

- 3 / 1 = - 3

9 / (-3) = - 3

-27 / (9) = - 3

=> ratio = - 3 means that every term is the previous one multiplied by - 3

3) terms

First term: 1 * (-3)^0 = 1

Second term: 1* (-3)^1 = - 3

Third term: 1 * (-3)^2 = 9

Fourth term: 1 * (-3)^3 = - 27

So, the summation is:

∞
∑ 1 * (-3)^n ,
n=0

which is read summation of one times negative three to the power of n from n equals zero to infinity => option C.

Problem 2. Write the sum using summation notation, assuming the suggested pattern continues.

- 4 + 5 + 14 + 23 + ... + 131

option  A) summation of the quantity negative four plus nine n from n equals zero to fifteen

Probe the statement A):

15
∑ (- 4 + 9n)
n=0

Now develop that summation: - 4 +9(0) - 4+ 9(1) - 4 + 9(2) - 4 + 9(3) ++ - 4 + 9(15) = - 4 + 5 + 14 + 23 + 131

Which is the very same sequence given.Therefore, the first statement if right.

Problem 3. Write the sum using summation notation, assuming the suggested pattern continues.
25 + 36 + 49 + 64 + ... + n^2 + ...

option A) summation of n squared from n equals five to infinity

Explanation

First term: 25 = 5^2
Second term: 36 = 6^2
Third term: 49 = 7^2
Fourth term: 64 = 8^2

nth term n^2

last term: the sequence is infinite

So, you can see that the sequence is the sum of the square of the integers from n = 5 to infinity =
∞
∑  (n^2)
n = 5

which is what summation of n squared from n equals five to infinity means.

Problem 4. Find the sum of the arithmetic sequence.
-1, 2, 5, 8, 11, 14, 17

option A) 56

Explanation:

You just have to sum all the terms (since they are few numbers that is the best way): - 1 + 2 + 5 + 8 + 11 + 14 + 17 = 56

56

Problem 5. Find the sum of the geometric sequence.
1, one divided by four, one divided by sixteen, one divided by sixty four, one divided by two hundred and fifty six

option D) 341 / 256

Explanation:
Write in form of fractions: 1 + 1/4 + 1/16 + 1/64 + 1/256

You'd better use the formula for the summation of a geometric sequence

k
∑  A * (r^n) = A * (1 - r^k) / (1 - r)
n=1

In this case: r = 1/4 (the ratio)
k = 5 (the number of terms)
A = 1 (the first term)

=> The sum = 1 * [1 - (1/4)^5 ] / [ 1 - 1/4], which when you simplify turns into 341 / 256 which is the option D.

Problem 6. Find the sum of the first 8 terms of the sequence. Show all work for full credit.
1, -3, -7, -11, ...

- 104

Explanation:

You can either sum the 8 terms or use the formula for the sum of an aritmetic sequence.

I will sum the 8 terms.

Note the the constant distance to sum is - 4, so the sum of the first eight terms is:

1  - 3 - 7 - 11 - 15 - 19 - 23 - 27 = - 104

The answer would be 27 because assuming that each package has 1 bar of soap and 1 tube of toothpaste, after 27 packages they would run out of toothpaste