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Chapter 12: Problem 44

Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$a=3, d=2, n=12$$

Short Answer

Expert verified
The partial sum \(S_{12}\) is 168.

Step by step solution

01

Identify the Arithmetic Sequence Formula

First, recognize that the sum of an arithmetic sequence can be found using the formula for the partial sum: \[ S_{n} = \frac{n}{2} \times (2a + (n-1)d) \] where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the number of terms.
02

Substitute Given Values

Next, substitute the given values \(a = 3\), \(d = 2\), and \(n = 12\) into the partial sum formula: \[ S_{12} = \frac{12}{2} \times (2 \times 3 + (12 - 1) \times 2) \].
03

Simplify the Expression

Simplify inside the parenthesis first: \(2 \times 3 = 6\) and \((12 - 1) \times 2 = 22\). So the expression becomes: \[ S_{12} = 6 \times (6 + 22) \].
04

Calculate the Final Result

Add the numbers inside the parentheses: \(6 + 22 = 28\). Then multiply by 6 to find \(S_{12}\): \[ S_{12} = 6 \times 28 = 168 \].

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Key Concepts

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

Partial Sum
The concept of a partial sum is essential when dealing with arithmetic sequences. It refers to the sum of the first few terms of a sequence, rather than the entire sequence itself.
When you calculate a partial sum, you're essentially adding up a certain number of terms from the start of the sequence.
  • The formula for the partial sum of an arithmetic sequence is: \[ S_{n} = \frac{n}{2} \times (2a + (n-1) \times d) \]
  • Here, \( n \) is the number of terms you want to sum, \( a \) is the first term, and \( d \) is the common difference between the terms.
Knowing how to find a partial sum helps when solving problems related to arithmetic sequences. It gives you control over specific segments of the sequence and can be especially useful in practical scenarios, such as budgeting over a specific timeframe.
Arithmetic Series
An arithmetic series is the addition of the terms of an arithmetic sequence. Each term in this sequence increases (or decreases) by a constant amount, known as the common difference.
The idea of a series comes into play when you add all these terms together.
  • The partial sum formula, mentioned earlier, is a way to calculate the sum of an arithmetic series up to a certain term.
  • An arithmetic series grows linearly, and each successive term is simply the previous term plus the common difference, \( d \).
For example, if you have an arithmetic sequence starting at 3 with a common difference of 2, the series would be 3, 5, 7, 9, and so on. Summing only the first 12 terms in this series, you'd use the partial sum formula to find the total sum efficiently, which was calculated to be 168 in the exercise.
Common Difference
The common difference in an arithmetic sequence is the consistent amount added to each term to get to the next. It's a crucial element that defines the nature of the sequence.
Understanding the common difference is vital because:
  • It helps in predicting future terms of the sequence quickly.
  • If you know the common difference \( d \) and the first term \( a \), you can write any term in the sequence as: \[ a_n = a + (n-1) \times d \] where \( a_n \) is the n-th term.
For instance, in the exercise, the common difference is 2. This means if the first term is 3, the second term would be 3+2=5, the third term 5+2=7, and so on. The consistency offered by the common difference is what allows for the use of formulas to find sums and specific terms efficiently.

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Most popular questions from this chapter

In this exercise we prove the identity $$\left(\begin{array}{c}n \\\r-1\end{array}\right)+\left(\begin{array}{l}n \\\r\end{array}\right)=\left(\begin{array}{c} n+1 \\\r\end{array}\right)$$ (a) Write the left-hand side of this equation as the sum of two fractions. (b) Show that a common denominator of the expression that you found in part (a) is \(r !(n-r+1) !\) (c) Add the two fractions using the common denominator in part (b), simplify the numerator, and note that the resulting expression is equal to the right- hand side of the equation.

Find the partial sum \(S_{n}\) of the geometric sequence that satisfies the given conditions. $$a_{2}=0.12, \quad a_{5}=0.00096, \quad n=4$$

Fibonacci's Rabbits Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair that becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the \(n\) th month? Show that the answer is \(F_{n},\) where \(F_{n}\) is the \(n\) th term of the Fibonacci sequence.

Find the 24 th term in the expansion of \((a+b)^{25}\).

The probability that an archer hits the target is \(p=0.9,\) so the probability that he misses the target is \(q=0.1 .\) It is known that in this situation the probability that the archer hits the target exactly \(r\) times in \(n\) attempts is given by the term containing \(p^{r}\) in the binomial expansion of \((p+q)^{n}\). Find the probability that the archer hits the target exactly three times in five attempts.
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