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Chapter 12: Problem 5

Write the first five terms of the sequence whose general term is given. \(a_{n}=2^{n}+3\)

Short Answer

Expert verified
The first five terms are 5, 7, 11, 19, and 35.

Step by step solution

01

Identify the General Term

The general term of the sequence is given by the formula: \(a_{n}=2^{n}+3\)
02

Calculate the First Term

To find the first term \(a_{1}\), substitute \(n=1\) into the general term formula: \(a_{1}=2^{1}+3=2+3=5\)
03

Calculate the Second Term

To find the second term \(a_{2}\), substitute \(n=2\) into the general term formula: \(a_{2}=2^{2}+3=4+3=7\)
04

Calculate the Third Term

To find the third term \(a_{3}\), substitute \(n=3\) into the general term formula: \(a_{3}=2^{3}+3=8+3=11\)
05

Calculate the Fourth Term

To find the fourth term \(a_{4}\), substitute \(n=4\) into the general term formula: \(a_{4}=2^{4}+3=16+3=19\)
06

Calculate the Fifth Term

To find the fifth term \(a_{5}\), substitute \(n=5\) into the general term formula: \(a_{5}=2^{5}+3=32+3=35\)

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Key Concepts

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

General Term
In sequences, the general term, often denoted as \(a_{n}\), is a formula that gives the value of terms depending on their position in the sequence.
Understanding the general term is crucial because it allows you to determine any term in the sequence without having to list all the previous terms.
For the given problem, the general term is \(a_{n}=2^{n}+3\).
Here, the general term combines an exponential function \(2^{n}\) and a constant +3.
To find any term, simply substitute the term number (n) into the formula. For example, for the third term, substitute 3 into the general term to get the value: \(a_{3}=2^{3}+3=11\).
Sequence Calculation
Calculating the terms in a sequence involves using the general term.
Here's how you can do it step-by-step:
  • Identify the general term formula. In this case, it is \(a_{n}=2^{n}+3\).
  • Substitute the position number (n) into the formula to get the value of the term.

For example:
  • To find the first term \(a_{1}\), substitute \(n=1\): \(a_{1}=2^{1}+3=5\)
  • For the second term \(a_{2}\), substitute \(n=2\): \(a_{2}=2^{2}+3=7\)
  • Continue this for as many terms as needed.
This method makes it easy to calculate any term in the sequence without listing all preceding terms.
Exponential Function
An exponential function is a mathematical expression where a constant base is raised to a variable exponent.
In our formula \(a_{n}=2^{n}+3\), \(2^{n}\) is the exponential function.
Exponential functions rapidly increase or decrease, depending on the base and the exponent.
Here the base is 2, which means each term doubles as the exponent \(n\) increases by 1.
  • For \(n=1\): \(2^{1}=2\)
  • For \(n=2\): \(2^{2}=4\)
  • For \(n=3\): \(2^{3}=8\)
  • And so on, the exponential function grows quickly.
Understanding exponential functions is essential as they often appear in algebra and other fields such as biology, physics, and economics.

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Most popular questions from this chapter

Expand each binomial using Pascal's Triangle. \((z-3)^{5}\)

Find the indicated term of a sequence where the first term and the common ratio is given. Find \(a_{10}\) given \(a_{1}=-6\) and \(r=-2\).

Find the coefficient of the indicated term in the expansion of the binomial. \(a^{4} b^{2}\) term of \((2 a+b)^{6}\)

Find the sum of the first 30 terms of each arithmetic sequence. \(-15,-12,-9,-6,-3, \ldots\)

Expand each binomial. \((2 x+5 y)^{4}\)
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