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Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblema 43
Capitolo 9, Problema 43

Use the formula for the sum of the first n terms of a geometric sequence to find the indicated sum.
∑i=165i\(\sum\)_{i=1}^{6} 5^i

Guida verificata passo dopo passo
1
Identify the geometric sequence and its parameters. Here, the terms are given by \( 5^i \) for \( i = 1 \) to \( 6 \). The first term \( a \) is \( 5^1 = 5 \), and the common ratio \( r \) is 5 because each term is multiplied by 5 to get the next term.
Recall the formula for the sum of the first \( n \) terms of a geometric sequence: \[ S_n = a \times \frac{r^n - 1}{r - 1} \] where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
Substitute the known values into the formula: \( a = 5 \), \( r = 5 \), and \( n = 6 \). This gives \[ S_6 = 5 \times \frac{5^6 - 1}{5 - 1} \].
Simplify the denominator \( 5 - 1 = 4 \) to rewrite the sum as \[ S_6 = 5 \times \frac{5^6 - 1}{4} \].
At this point, you can calculate \( 5^6 \), subtract 1, then multiply by 5 and divide by 4 to find the sum of the series.

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Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. For example, in the sequence 5, 25, 125, ..., each term is multiplied by 5. Understanding this helps identify the pattern and apply the correct formulas.
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Geometric Sequences - Recursive Formula

Sum of the First n Terms of a Geometric Sequence

The sum of the first n terms of a geometric sequence can be calculated using the formula S_n = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio. This formula efficiently computes the total without adding each term individually.
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Geometric Sequences - Recursive Formula

Sigma Notation

Sigma notation (∑) is a concise way to represent the sum of a sequence of terms. It specifies the index of summation, the starting and ending values, and the general term formula. Understanding sigma notation helps translate the problem into a sum that can be evaluated using formulas.
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Interval Notation
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