#arithmetic series

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Arithmetic Progression

Arithmetic Progression (A.P) is a sequence of numbers where difference between the two consecutive numbers is same. For example, In an Arithmetic Progression: 1, 4, 7, 10, 13, ….. the difference between the all two consecutive numbers is same. 4 -1 = 3 7 – 4 = 3 10 – 7 = 3 and so on Arithmetic Progression is also called Arithmetic Sequence. nth term of an Arithmetic Progression If a1 is the first…

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Arithmetic Series

An arithmetic series is transforming an arithmetic sequence into an equation so that it shows the importance of its sum. Arithmetic sequence: 1, 2, 3, 4, ... Arithmetic series: 1 + 2 + 3 + 4 + ... Just like arithmetic sequences, arithmetic series have infinite arithmetic series. Arithmetic series: 1 + 2 + 3 + 4 Infinite arithmetic series: 1 + 2 + 3 + 4 + ... Arithmetic Series, Sn

Sn = the sum of the first n terms, said as "sum of first n terms" tn = general term (mostly the "imaginary" last term of the sequence), said as "nth term" t₁ = first term d = common difference n = number of terms in the series Finding the Sum of an Arithmetic Series 1. Determine the appropriate arithmetic series formula to use by considering the variables given. If the question provides information on Sn and tn, but the question asks for Sₐ, use the second formula to solve for Sₐ. If the question provides an arithmetic series with a known common difference, d, and a known first term, t₁, use the second formula to determine Sn. After determining the appropriate formula, set up this formula with respect to the sum needed to be found. 2. Find the common difference, d, using the arithmetic series and common difference formula, and find the first term, t₁, by using the initial term in the arithmetic series. 2.1 If the common difference, d, or the first term, t₁, are unable to be found, use the first arithmetic series formula with the given tn and Sn value to solve for t₁, or use the second arithmetic series formula to solve for the common difference, d. 3. With all variables found, substitute them into the arithmetic series formula and solve for the missing sum value.

PRACTICE QUESTIONS (BOOKLET) Page: 19 #4, 6, 7, 10

Given the following infinite arithmetic series: -75 - 69 - 63 - 57 - 51 - 45 - ... Determine S₆. 1. Determine the appropriate arithmetic series formula to use by considering the variables given. If the question provides information on Sn and tn, but the question asks for Sₐ, use the second formula to solve for Sₐ. If the question provides an arithmetic series with a known common difference, d, and a known first term, t₁, use the second formula to determine Sn. After determining the appropriate formula, set up this formula with respect to the sum needed to be found. Since we are given an arithmetic series with a known common difference, d, and a known first term, t₁, we will use the second arithmetic series formula. And, since the sum needed to be found is S₆, we will set up the formula with respect to S₆. Setting up the formula, where Sn = S₆, n = 6:

2. Find the common difference, d, using the arithmetic series and common difference formula, and find the first term, t₁, by using the initial term in the arithmetic series. Finding the common difference, d, where t₂ = -69, t₁ = -75: d = t₂ - t₁ d = -69 - (-75) d = -69 + 75 d = 6 Finding the first term, t₁, by looking at the arithmetic series and seeing the initial term: t₁ = -75 3. With all variables found, substitute them into the arithmetic series formula and solve for the missing sum value. Substituting values, where d = 6, t₁ = -75:

Therefore, the sum of the first 6 terms is -360, or S₆ = -360.

Given the following information: S15 = 93.75, d = 0.75, and t15 = 11.5 Determine S₃. 1. Determine the appropriate arithmetic series formula to use by considering the variables given. If the question provides information on Sn and tn, but the question asks for Sₐ, use the second formula to solve for Sₐ. If the question provides an arithmetic series with a known common difference, d, and a known first term, t₁, use the second formula to determine Sn. After determining the appropriate formula, set up this formula with respect to the sum needed to be found. Since S15 and t15 do not relate with S₃, we will use the second formula with respect to S₃. Setting up the formula, where Sn = S₃, n = 3, d = 0.75:

2. Find the common difference, d, using the arithmetic series and common difference formula, and find the first term, t₁, by using the initial term in the arithmetic series. There is no way of finding the first term, t₁, with the given information, so we skip to step 2.1. 2.1 If the common difference, d, or the first term, t₁, are unable to be found, use the first arithmetic series formula with the given tn and Sn value to solve for t₁, or use the second arithmetic series formula to solve for the common difference, d. We will use the second arithmetic series formula to find the first term, t₁. Setting up the formula, where Sn = S15 = 93.75, tn = t15 = 11.5, n = 15:

Solving for the first term, t₁:

3. With all variables found, substitute them into the arithmetic series formula and solve for the missing sum value. Substituting values, where t₁ = 1:

Therefore, the sum of the first 3 terms is 5.25, or S₃ = 5.25.

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Arithmetic Series

Arithmetic series: a sum of terms that form an arithmetic sequence. Example: For the arithmetic sequence 2, 4, 6, 8 the arithmetic series is represented by 2 + 4 + 6 + 8. Now there are two formulas for solving the sum of an arithmetic series. Formula one used if you have the final term's value:

Formula two used otherwise:

A particular firefly flashes twice in the first minute, four times in the second minute, and six times in the third minute. If this pattern continues, what is the number of flashes in the 30th minute? What is the total number of flashes in 30 minutes? Information given: t1 = 2, n = 30, d = 2 The first question deals with a sequence formula, so let us solve this one by plugging the given information with the sequence formula.

Next, we use the series formula for finding the sum of all those flashes within 30 minutes. Since we know the 30th term's value, we can use the first formula.

The sum of the first two terms of an arithmetic series is 13 and the sum of the first four terms is 46. Determine the sixth term of the series and the sum to six terms. Information given: t2 = 13, t4 = 46 Another good thing we should know is the common difference but it is not shown or easily calculated from the problem. So we need to develop a system of equations with the two general terms of t2 and t4. We will use the second series formula. For t2:

For t4:

Now we put them together (whichever order) and subtract then solve for d.

By the way, t1 cancels each other out. Now that we know the common difference, we plug it in to one of the system of equations and solve for t1.

Now that we know the values for t1 and d, we create a general term with the sequence formula and series formula. General term for sequence formula:

General term for series formula (second one since we do not know the 6th term's value):

Next, we plug in 6 for both general terms and solve! 6th term:

Sum of first 6 terms: