Skip to main content
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 57
Capitolo 9, Problema 57

Express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. a+ar+ar2+⋯+ ar12

Guida verificata passo dopo passo
1
Identify the pattern of the terms in the sum: the first term is \( a \), the second term is \( ar \), the third term is \( ar^2 \), and so on, up to \( ar^{12} \).
Recognize that this is a geometric series where each term is obtained by multiplying the previous term by \( r \). The general term can be written as \( ar^k \) where \( k \) is the exponent of \( r \).
Choose the index of summation \( k \) to start at 0, since the first term corresponds to \( r^0 = 1 \). So, the lower limit of summation is \( k = 0 \).
Determine the upper limit of summation. Since the last term is \( ar^{12} \), the upper limit is \( k = 12 \).
Write the sum in summation notation as: \[ \sum_{k=0}^{12} a r^k \]

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Summation Notation

Summation notation is a concise way to represent the sum of a sequence of terms using the sigma symbol (∑). It includes an index of summation, lower and upper limits, and a general term formula. This notation simplifies writing long sums and helps in analyzing series.
Video consigliato:
Percorso guidato
05:18
Interval Notation

Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio r. The general form is a + ar + ar² + ... + ar^n. Recognizing this pattern helps in expressing the sum compactly and finding closed-form formulas.
Video consigliato:
Percorso guidato
4:18
Geometric Sequences - Recursive Formula

Index of Summation and Limits

The index of summation (commonly k) represents the variable that changes in each term of the sum. The lower and upper limits define the starting and ending values of this index. Choosing appropriate limits is essential for accurately representing the sum in summation notation.
Video consigliato:
Percorso guidato
03:50
Adding & Subtracting Like Radicals
Pratica correlata
Domanda del libro di testo

In Exercises 55–60, express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. a+(a+d)+(a+2d)+⋯+(a+nd)a+(a+d)+(a+2d)+⋯+ (a+nd)

914
views
Domanda del libro di testo

Use mathematical induction to prove that the statement is true for every positive integer n. 1 + 4 + 4^2 + ... + 4^(n-1) = ((4^n)-1)/3

802
views
Domanda del libro di testo

Find the middle term in the expansion of (3/x + x/3)10

593
views
Domanda del libro di testo

Let {an} = - 5, 10, - 20, 40, ..., {bn} = 10, - 5, - 20, - 35, ..., {cn} = - 2, 1, - 1/2, 1/4 Find a10 + b10.

847
views
Domanda del libro di testo

Use mathematical induction to prove that the statement is true for every positive integer n. 5 + 10 + 15 + ... + 5n = (5n(n+1))/2

1214
views
Domanda del libro di testo

Let {an} = - 5, 10, - 20, 40, ..., {bn} = 10, - 5, - 20, - 35, ..., {cn} = - 2, 1, - 1/2, 1/4 Find the difference between the sum of the first 10 terms of {an} and the sum of the first 10 terms of {bn}.

791
views