I know my answer, but I stumbled on it accidentally. I want to know exactly HOW to get my answer. The question states: In a geometric sequence, a1=12 and r=(sqrt)2. What is the approximate sum of the first 20 terms of the sequence?
why worry about the approximate sum? Use your formula to get
S20 = 12((√2)^20-1)/(√2-1) = 12(1024-1)/(√2-1) = 12276/(√2-1)
What's my formula? I don't know it.
If it is not in your text, burn your text.
https://www.varsitytutors.com/hotmath/hotmath_help/topics/sum-of-the-first-n-terms-of-a-geometric-sequence
To find the approximate sum of the first 20 terms of a geometric sequence, we can use the formula for the sum of a geometric series. The formula is given as:
Sn = a1 * (1 - r^n) / (1 - r),
where Sn represents the sum of the first n terms, a1 is the first term, r is the common ratio of the sequence, and n is the number of terms.
In this case, a1 = 12, r = √2, and n = 20. Now we can substitute these values into the formula to find the approximate sum:
Sn = 12 * (1 - (√2)^20) / (1 - √2).
Evaluating (√2)^20, we get approximately 64. Combining the terms, we have:
Sn ≈ 12 * (1 - 64) / (1 - √2).
Next, we simplify the expression further:
Sn ≈ 12 * (-63) / (1 - √2).
Finally, we can calculate the approximate sum:
Sn ≈ -756 / (1 - √2).
Using a calculator, we can find that the approximate value of Sn is approximately -2,720.12. Therefore, the approximate sum of the first 20 terms of the geometric sequence is -2,720.12.