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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 29
Capitolo 9, Problema 29

Find each indicated sum. ∑i=165i\(\sum\)_{i=1}^{6} 5i

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Identify the summation notation given: \(\sum_{i=1}^{6} 5i\). This means you need to find the sum of the expression \$5i$ as $i$ goes from 1 to 6.
Rewrite the summation by expanding the terms: \(5(1) + 5(2) + 5(3) + 5(4) + 5(5) + 5(6)\).
Factor out the constant 5 from the summation: \(5 \sum_{i=1}^{6} i\).
Use the formula for the sum of the first \(n\) natural numbers: \(\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\). Here, \(n=6\).
Substitute \(n=6\) into the formula and multiply by 5: \(5 \times \frac{6 \times (6+1)}{2}\). This expression represents the sum you need to calculate.

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Summation Notation (Sigma Notation)

Summation notation uses the Greek letter sigma (Σ) to represent the sum of a sequence of terms. It specifies the index of summation, the lower and upper limits, and the expression to be summed. For example, Σ from i=1 to 6 of 5i means adding 5 times each integer from 1 to 6.
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Interval Notation

Arithmetic Series

An arithmetic series is the sum of terms in an arithmetic sequence, where each term increases by a constant difference. In this problem, 5i forms an arithmetic sequence with a common difference of 5. Understanding this helps in applying formulas to find the sum efficiently.
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Arithmetic Sequences - General Formula

Formula for the Sum of an Arithmetic Series

The sum of the first n terms of an arithmetic series can be found using the formula S_n = n/2 (first term + last term). This formula simplifies the calculation by avoiding term-by-term addition, making it useful for quickly finding sums like Σ 5i from i=1 to 6.
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Arithmetic Sequences - General Formula
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