Question

6 Evaluate the series 8Σn=3 5n. What is the lower limit of the summation natation is n=3
125
38
210
165

Answer 1

165

Answer 2

how did you get that?

Answer 3

To get the sum, we substitute the values of n into the expression 5n and add them up as follows:

8Σn=3 5n = 5(3) + 5(4) + 5(5) + ... + 5(10)

= 15 + 20 + 25 + ... + 50

We can see that this is an arithmetic series with first term a=15, common difference d=5, and last term l=50. We can use the formula for the sum of an arithmetic series:

Sn = (n/2)(a + l)

where n is the number of terms in the series. Since we are summing from n=3 to n=10, the number of terms is:

n = 10 - 3 + 1 = 8

Substituting the values, we get:

S8 = (8/2)(15 + 50) = 8(65) = 520

Therefore, the answer is 520 and the lower limit of the summation notation is n=3.

Answer 4

Show your work
6 Evaluate the series 8Σn=3 5n. What is the lower limit of the summation natation is n=3
125
38
210
165

9. What is the sum of the geometric series 10Σn=1 6(2)^n
756
12,276
6,138

Answer 5

The first term of the geometric series is:

a = 6(2)^1 = 12

The common ratio is:

r = 2

We can use the formula for the sum of a geometric series:

Sn = a(1 - r^n) / (1 - r)

Substituting the values, we get:

10Σn=1 6(2)^n = 6(2)^1 + 6(2)^2 + ... + 6(2)^10

= 12 + 24 + 48 + ... + 12,288

To find the number of terms in the series, we can use the formula:

l = a * r^(n-1)

where l is the last term of the series. Solving for n, we get:

n = log(l/a) / log(r) + 1

Substituting the values, we get:

n = log(12,288/12) / log(2) + 1 = 11

Therefore, the number of terms in the series is:

n = 11 - 1 + 1 = 11

Substituting again, we get:

Sn = 12(1 - 2^11) / (1 - 2) = 12(1 - 2048) / (-1) = 12(2047) = 24,564

Therefore, the sum of the geometric series is 24,564.

Answer 6

As usual, the bot messed up in the explanation
There are 6 terms, not 8, the last term is 40 , not 50
and then comes up with the wrong answer of course, even though
it did get the right answer of 165 for its first response.

For the first one , which the bot also messed up
you want Σ 5n , where n goes from 3 to 8
= 15 + 20 + 25 + 30 + 35 + 40 = (6/3)(15 + 40) = 165

for the second: Σ 6(2)^n , from 1 to 10
= 6(2^1) + 6(2^2) + .... + 6(2^10)
this is a GP with a = 12, r = 2 and we have 10 terms
sum(1) = a(r^10 - 1)/(2-1)
= 12(2^10 - 1)/1 = 12276

Answer 7

I apologize for the mistakes in my previous responses. Thank you for providing the correct solutions.