# Sum of 5^1-5^2+5^3-5^4+...-5^{98}: e. (5/6)(1-5^98)

• Biosyn
In summary, the sum of the given series is equal to (5/6)(1-(-5)^98). This can be calculated using the formula Sn = (a1*(1-r^n))/(1-r), where a1 is the first term, r is the common ratio, and n is the number of terms. In this case, a1 = 5, r = -5, and n = 98.
Biosyn

## Homework Statement

Find the sum of $5^1-5^2+5^3-5^4+...-5^{98}$

a. (5/4)(1-5^99)
b. (1/6)(1-5^99)
c. (6/5)(1+5^98)
d. (1-5^100)
e. (5/6)(1-5^98)

## The Attempt at a Solution

I feel as though this is actually a simple problem and that I'm not looking at it the right way.

[$5^1 + 5^3 + 5^5...5^{97}$] + [$-5^2-5^4-5^6...-5^{98}$]

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Do you know how to sum ##x^n## in general? What is ##x## here?

jbunniii said:
Do you know how to sum ##x^n## in general? What is ##x## here?

##x## will be 5?

$$\sum_{i=0}^{48} (5^{2i + 1})$$ + $$\sum_{i=0}^{49} (5^{2i})$$Never mind, I figured it out!

Last edited:
Actually, it looks to me like
$$-\sum_{n=1}^{98}(-5)^n$$

jbunniii said:
Actually, it looks to me like
$$-\sum_{n=1}^{98}(-5)^n$$

I used Sn = $\frac{a_1*(1-r^n)}{1-r}$

Sn = $\frac{5*(1-(-5)^98)}{1-(-5)}$

= $\frac{5*(1-(-5)^98)}{6}$

= (5/6)*(1-(-5)^98)

## 1. What is the pattern in the sum of the given series?

The given series follows a pattern of alternating addition and subtraction of powers of 5, starting with 5^1 and ending with 5^{98}. This creates a geometric series with a common ratio of -5.

## 2. How do you find the sum of the given series?

To find the sum of the given 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 first n terms, a is the first term, and r is the common ratio. In this case, a = 5^1 = 5, r = -5, and n = 98. Plugging in these values, we get Sn = (5(1-(-5)^98))/(1-(-5)) = (5/6)(1-5^98).

## 3. What is the significance of the value (5/6)(1-5^98) in the sum of the given series?

The value (5/6)(1-5^98) represents the sum of the first 98 terms of the given series. It is the finite sum of the series and is equal to -1.066666667 x 10^97.

## 4. Can the given series be simplified?

Yes, the given series can be simplified by using the formula for the sum of a geometric series. Simplifying the sum gives us (5/6)(1-5^98), which is the finite sum of the series.

## 5. What is the limit of the given series as n approaches infinity?

The limit of the given series as n approaches infinity is undefined. This is because the series is a geometric series with a common ratio of -5, which is greater than 1 in absolute value. This means that the series does not converge and the sum of the infinite terms is undefined.

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