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Chapter 11: Problem 112

Find \(a_{5}\) if \(a_{1}=-6\) and \(a_{n}=2 a_{n-1}+3\) for \(n>1\).

Short Answer

Expert verified
-51

Step by step solution

01

Identify the terms

We know that the first term of the sequence is given by: a_{1} = -6.
02

Apply the recurrence relation for n = 2

Using the recurrence relation, calculate the second term: a_{2} = 2a_{1} + 3 = 2(-6) + 3 = -12 + 3 = -9.
03

Apply the recurrence relation for n = 3

Using the recurrence relation, calculate the third term: a_{3} = 2a_{2} + 3 = 2(-9) + 3 = -18 + 3 = -15.
04

Apply the recurrence relation for n = 4

Using the recurrence relation, calculate the fourth term: a_{4} = 2a_{3} + 3 = 2(-15) + 3 = -30 + 3 = -27.
05

Apply the recurrence relation for n = 5

Using the recurrence relation, calculate the fifth term: a_{5} = 2a_{4} + 3 = 2(-27) + 3 = -54 + 3 = -51.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Recurrence Relation
A recurrence relation is a mathematical way of defining a sequence where each term is formulated as a function of its preceding terms. This approach is particularly useful in identifying complex sequences without directly calculating every single term.
In our exercise, the recurrence relation is: \( a_n = 2a_{n-1} + 3 \)
This means that each term in the sequence can be determined by multiplying the previous term by 2 and then adding 3.
Understanding this relation forms the basis of solving the problem step by step.
For instance:
  • To find \(a_2\), we use \(a_1\) as given: \(a_2 = 2a_1 + 3\).
  • Next, for \(a_3\), we use \(a_2\) that we just calculated: \(a_3 = 2a_2 + 3\).
Recurrence relations help break down complex problems into simpler, manageable steps, easing calculations and promoting better understanding.
Sequence Calculation
Calculating terms in a sequence using a recurrence relation involves applying the relation recursively until the desired term is found.
Let's see how this works in our given exercise:
Given:\( a_1 = -6 \)
We need to find \( a_5 \) using the recurrence relation: \( a_n = 2a_{n-1} + 3 \)
  • Step 1: Start with \( a_1 = -6 \).
  • Step 2: Calculate \( a_2 \): \( a_2 = 2(-6) + 3 = -12 + 3 = -9 \).
  • Step 3: Calculate \( a_3 \): \( a_3 = 2(-9) + 3 = -18 + 3 = -15 \).
  • Step 4: Calculate \( a_4 \): \( a_4 = 2(-15) + 3 = -30 + 3 = -27 \).
  • Step 5: Calculate \( a_5 \): \( a_5 = 2(-27) + 3 = -54 + 3 = -51 \).
By systematically applying the recurrence relation at each step, we were able to determine \( a_5 \). This methodical approach ensures that no terms are missed and that the correct term is identified.
Term Identification
Identifying specific terms in a sequence defined by a recurrence relation is crucial, and it requires consistency in applying the initial conditions and the formula.
In our specific problem, we begin by identifying \( a_1 \) which is provided as \( -6 \). Every subsequent term is a direct application of the recurrence relation:\( a_n = 2a_{n-1} + 3 \)
When calculating a term, ensure to use the correct preceding term:
  • Knowing \( a_1 \), we find \( a_2 = -9 \).
  • Using \( a_2 \), we find \( a_3 = -15 \).
  • Building on \( a_3 \), we determine \( a_4 = -27 \).
  • Finally, using \( a_4 \), we calculate \( a_5 = -51 \).
This process ensures that each step adheres to the defined relation, thereby accurately identifying the requested term. Properly identifying terms is essential to solving any recurrence-based sequence problem accurately and methodically.

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