evaluate each geometric series described
21) -3 + 15 - 75 + 375..., n=6
22) 1 + 5 + 25 + 125..., n=8
r = -5
r = 5
You can tell what a is, and now
Sn = a(r^n - 1)/(r-1)
so just plug in your numbers
21) Ah, the classic "growing negativity" series! Let's evaluate this funny geometric series. Starting with a term of -3, each term is obtained by multiplying the previous term by -5. So, to find the 6th term, let me just grab my multiplying shoes...-3 * -5 * -5 * -5 * -5 * -5 = 375. Voila! The 6th term is 375.
22) Looks like we have a "powers of 5" series here! Starting with a term of 1, each term is obtained by multiplying the previous term by 5. Let's see how big this series gets. 1 * 5 * 5 * 5 * 5 * 5 * 5 * 5 = 976,562. Oof! Looks like this series has really grown. The 8th term is 976,562.
To evaluate each geometric series, we can use the formula for the sum of a geometric series:
Sn = a(1 - r^n) / (1 - r)
where:
- Sn is the sum of the series
- a is the first term of the series
- r is the common ratio of the series
- n is the number of terms
Let's calculate each one separately:
21) -3 + 15 - 75 + 375... with n = 6
In this series, the first term (a) is -3 and the common ratio (r) is 5.
Using the formula:
Sn = -3(1 - 5^6) / (1 - 5)
= -3(1 - 15625) / -4
= -3(-15624) / -4
= 46872 / 4
= 11718
Therefore, the sum of the series with n = 6 is 11718.
22) 1 + 5 + 25 + 125... with n = 8
In this series, the first term (a) is 1 and the common ratio (r) is 5.
Using the formula:
Sn = 1(1 - 5^8) / (1 - 5)
= 1(1 - 390625) / -4
= 1(-390624) / -4
= -390624 / -4
= 97656
Therefore, the sum of the series with n = 8 is 97656.
To evaluate a geometric series, we use the formula:
Sn = a * (1 - r^n) / (1 - r)
where Sn is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.
Let's calculate each of the geometric series.
21) -3 + 15 - 75 + 375..., n=6
In this series, the first term (a) is -3, and the common ratio (r) can be found by dividing any term by its previous term. So, we divide 15 by -3 to get the common ratio:
r = 15 / -3 = -5
Now, we can substitute these values into the formula:
Sn = -3 * (1 - (-5)^6) / (1 - (-5))
Simplifying the equation, we have:
Sn = -3 * (1 - 15625) / 6
Sn = -3 * (-15624) / 6
Sn = 46872
Therefore, the sum of the geometric series with n=6 is 46872.
22) 1 + 5 + 25 + 125..., n=8
In this series, the first term (a) is 1, and the common ratio (r) can be found by dividing any term by its previous term. So, we divide 5 by 1 to get the common ratio:
r = 5 / 1 = 5
Now, we can substitute these values into the formula:
Sn = 1 * (1 - 5^8) / (1 - 5)
Simplifying the equation, we have:
Sn = 1 * (1 - 390625) / 4
Sn = 1 * (-390624) / 4
Sn = -97656
Therefore, the sum of the geometric series with n=8 is -97656.