If the sum of an infinite geometric series is 20 and the common ratio is 1/4, wh

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Question: If the sum of an infinite geometric series is 20 and the common ratio is 1/4, what is the first term?

Options:

Correct Answer: 25

Solution:

The sum S of an infinite GP is given by S = a / (1 - r). Here, 20 = a / (1 - 1/4) = a / (3/4). Thus, a = 20 * (3/4) = 15.

If the sum of an infinite geometric series is 20 and the common ratio is 1/4, what is the first term?

If the sum of an infinite geometric series is 20 and the common ratio is 1/4, what is the first term?

Step 1: Understand that we have an infinite geometric series with a sum S = 20 and a common ratio r = 1/4.
Step 2: Recall the formula for the sum of an infinite geometric series: S = a / (1 - r), where a is the first term.
Step 3: Substitute the known values into the formula: 20 = a / (1 - 1/4).
Step 4: Calculate 1 - 1/4, which equals 3/4.
Step 5: Now the equation looks like this: 20 = a / (3/4).
Step 6: To solve for a, multiply both sides of the equation by 3/4: a = 20 * (3/4).
Step 7: Calculate 20 * (3/4), which equals 15.
Step 8: Therefore, the first term a is 15.

Infinite Geometric Series – The sum of an infinite geometric series is calculated using the formula S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.
Common Ratio – Understanding the significance of the common ratio in determining the convergence of the series and its impact on the sum.
Algebraic Manipulation – The ability to rearrange and solve equations to isolate the variable of interest, in this case, the first term 'a'.