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Problem Solving with Series Sums

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Katrina just started her new job last week. She is making $60 per day, and saving 10%. Her savings account has been slowly building up at$6 per day, and currently has \$73 in it.

How much money will she have after working another 15 days? How much after another 27 days?

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Guidance

In the previous lesson, we discussed Gauss' formula for finding the total of the numbers in a series. In this lesson, we continue to practice that, and also to make use of the modified version of Gauss' Formula that is commonly seen in other texts:

The sum of the first n terms in an arithmetic series is $S_n = \frac{n(a_1 + a_n)} {2}$

Since this lesson is mostly about applying the rules for series, there is minimal instruction. If you would like a more in-depth presentation on series, please see the prior lesson: "Series Sums and Gauss' Formula."

Example A

Find the sum of the first 50 terms of an arithmetic series if the first term is 5 and the common difference is 3.

Solution

$S_{50} = \frac{50(a_1 + a_n)} {2}$ ..... Substitute the applicable number of terms in the sequence
$\frac{50(5 + (5 + 49 \times 3))} {2}$ ..... Substitute the values for the first and 50th terms
$\frac{50(157)} {2} = (25)(127)$ ..... Simplify
$3,175$

This method is clearly much easier than writing out and adding 50 numbers!

Example B

Find the sum of the first 40 terms of an arithmetic series in which the first term is 8 and the common difference is 5.

Solution

Use the formula $S_n = \frac{n(a_1 + a_n)} {2}$

$S_{40} = \frac{40(8 + 39\cdot5)} {2}$ ..... Substitute in the number of terms, and the value of the first and last terms
$S_{40} = \frac{40(8 + 195)}{2}$ ..... Simplify
$S_{40} = \frac{8120}{2}$ ..... Simplify
$S_{40} = 4060$

Example C

Consider the series: -13 + -3 + 7 + 17 + 27 + ...

a) What is the 25 th term?
b) What is the sum of the first 25 terms?

Solution

a) This is an arithmetic series where each successive term is 10 more than the last
$-13 + 24(10) = 227$
b) Use the formula $S_n = \frac{n(a_1 + a_n)} {2}$
This is the sum of the first 25 terms of an arithmetic series with a common difference of 10
$S_{25} = \frac{25(-13 + 227)} {2}$ ..... Substitute in the number of terms, and the value of the first and last terms
$S_{25} = \frac{25 \cdot 214}{2}$ ..... Simplify
$S_{25} = \frac{5350}{2}$ ..... Simplify
$S_{25} = 2675$

Vocabulary

A hypothesis is a conclusion made on the basis of evidence, or a statement assumed to be true for the sake of an argument.

Mathematical induction is a method of mathematical proof used to establish that a given statement is true of all positive integers (natural numbers).

A partial sum is the sum of the first n terms in an infinite series, where n is some positive integer.

Guided Practice

Questions

1) A student needed to find the sum of the first 10 terms of the series 4 + 12 + 36 + ... and so he wrote the following:

$S_{10} = \frac{10(4 + 4 \times 3^9)} {2} = \frac{10(78,736)} {2} = 393,680$
Do you agree with the student’s work? Explain.

2) Given an arithmetic sequence $(a_n)$ determined by $a_1 = 143$ and $d = -3$

What is the 220th number of the sequence.
We find the solution by using the following formula:
$a_n = a + (n-1)d$

3) How do we find the sum of the 220th number of the sequence, given the same information as the 2nd guided practice above?

We use the following formula: $s_n = \frac{a_1 + a_n}{2}(n)$ . We set n = 220.

Solutions

1) Because the series is geometric, this formula is not appropriate. The work here does not represent the sum of the first 10 terms. Using a graphing calculator, you can find that the sum is 118,096.

2) To find the 220th number:

$a_{220} = a_1 + 219d$
$= 143 + 219 (-3)$
$= -514$

3) To find the sum of the first 220 numbers:

$s_{220} = \frac{a_1 + a_{220}}{2}(220)$
$s_{220} = \frac{143 + (-514)}{2}(220)$
$s_220 = \frac{-371}{2}\cdot 220$
the $220^{th}$ number of the sequence is = $-40,810$

Practice

Use the arithmetic formula to solve the following problems:

1. Given the arithmetic series of numbers: 1, 4, 7, 10, 13... a)Find the 200th number in the sequence b)find the sum of the first 200 numbers.
2. The sum of the first five numbers of an arithmetic sequence is -45. What is the value of the third number? ( hint: find $a_3$ if $s_5 = -45$ )
3. You have an arithmetic series of numbers defined by: $a_1 = 45$ and $d= -5$ a)Determine $a_150$ b)Identify the sum of: $a_1 + a_2 +... + a_{150}$
4. The sum of the first three numbers in an arithmetic sequence is 219. The sum of the first nine numbers in the same sequence is 603. What is the 143rd number of the sequence.
5. The first eight numbers of an arithmetic sequence add up to 604. The next eight numbers added up equal 156. Find the first number and the common difference in the sequence.
6. The first number in an arithmetic sequence is 80. Find the common difference if we also know that $s_9$ is eighteen times $a_{11}$
7. If $a_n$ is an arithmetic sequence with $a_1 = 1$ . Find the second number if we know that the sum of the first five numbers is one-fourth of the sum of the next five numbers.
8. Given $(a_n) = 78, 75, 72, 69...$ Find $a_{150}$ and $s_{150}$
9. The following conditions exist within a sequence of numbers: $a_{50} = 252$ and $s_{50} = 2800$ What is the first number of the series, and what is the common difference?
10. What are the values of $a$ and $d$ , given that $(a_n)$ is an arithmetic series of numbers, if we know: $a_{15} = 62$ and $s_{20} = 700$ ?
11. Given the sequence: $a_1 = -16$ and $d = \frac{1}{3}$ Find the values of n, so that $s_n = 50$
12. If $a_1 = 8$ and $d = -3$ What are the values of $a_{20}$ and $s_{20}$
13. If $a_{34} = 193$ and $s_{17} = 306$ find $a$ and $d$
14. Three sides of a right triangle are integers and form an arithmetic sequence. Find the sides of the triangle