Problem 91
Question
Explain how to find the sum of the first \(n\) terms of a geometric sequence without having to add up all the terms.
Step-by-Step Solution
Verified Answer
Without adding all terms, the sum of the first \(n\) terms of a geometric sequence can be found by substituting the terms into the formula \(S_n = a (1 - r^n) / (1 - r)\) and calculating the result.
1Step 1: Identify the parameters of the geometric sequence
Identify the first term \(a\) of the sequence, the common ratio \(r\), and the number of terms \(n\) which you want to find the sum of.
2Step 2: Substitute into the formula
Substitute the values for \(a\), \(r\) and \(n\) into the formula \(S_n = a (1 - r^n) / (1 - r)\).
3Step 3: Solve the equation
After substitution of the values, solve the equation to get the sum of the first \(n\) terms of the geometric sequence.
Other exercises in this chapter
Problem 90
Evaluate \(\frac{n !}{(n-r) !}\) for \(n=20\) and \(r=3\)
View solution Problem 90
What appears to be happening to the terms of each sequence as n gets larger? $$ a_{n}=\frac{100}{n} \quad n:[0,1000,100] \text { by } a_{n}:[0,1,0.1] $$
View solution Problem 91
Will help you prepare for the material covered in the next section. Use the formula \(a_{n}=a_{1} 3^{n-1}\) to find the seventh term of the sequence \(11,33,99,
View solution Problem 91
Evaluate \(\frac{n !}{(n-r) ! r !}\) for \(n=8\) and \(r=3\)
View solution