Respuesta :
6.
1, -3, -7, -11, -15, -19, -23, -27...
sum:
8/2 (1 + -27)
8/2 (-26)
-104
1, -3, -7, -11, -15, -19, -23, -27...
sum:
8/2 (1 + -27)
8/2 (-26)
-104
This are 6 question and 6 answers
Problem 1. Write the sum using summation notation, assuming the suggested pattern continues.
1 - 3 + 9 - 27 + ...
Answer: 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
Answer: 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 + ...
Answer: 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
Answer: 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
Answer: 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
Answer: 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, ...
Answer: - 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
Problem 1. Write the sum using summation notation, assuming the suggested pattern continues.
1 - 3 + 9 - 27 + ...
Answer: 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
Answer: 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 + ...
Answer: 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
Answer: 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
Answer: 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
Answer: 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, ...
Answer: - 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
