Arithmetic Sequence Series Problem

In summary, we can use the fact that terms in an arithmetic sequence vary by a constant to solve for x in the first problem, find the sums of the first twelve and twenty-four terms in the second problem and then use this information to find the sum of the first twenty-eight terms. In the third problem, we can use the given terms to find the values of a and d, which will allow us to find any term tn in the sequence.
  • #1
nobb
33
0
Hi. Please explain to me how to do these three problems:

1. The terms x+3, 3x-1, and 7x-2 are consecutive terms in an arithmetic sequence. Find x.

2. The sum of the first twenty-eight terms of an arithmetic series if the sum of the first twelve terms is -72 and the sum of the first twenty-four terms is 144.

3. Find a, d, and tn for this sequence: t4= -9 , t15 = -31
 
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  • #2
The terms in an arithmetic sequence always vary by a constant:
a+ r, a+ 2r, a+ 3r, etc so that subtracting and consecutive two terms gives r.

1. Here we must have (3x-1)- (x+1)= r= (7x-2)- (3x-1). Ignore the r and solve for x.

2. With an arithmetic sequence you can find the "average" of all numbers by averaging the first and last terms of the sequence. The sum of n terms is just n times that average.

In particular the average of the first 12 terms is (a1+ a12)/2 so the sum of the first 12 terms is 12(a1+ a12)/2= 6(a1+ a12)= -72.

Similarly, the sum of the first 24 terms is 12(a1+ a24)= 144.

Of course, if we take a1 as the first term and d as the common difference, a12= a1+ 11d and a24= a1+23d.
Putting those into the two equations for the sums gives you 2 equations in the two unkowns, d and a1. Once you know those, you can calculate a28= a1+ 27d and the sum is 28(a1+ a28[/sub)/2= 14(a1+ a28).

a1= a and ann= a+ (n-1)d so t4 (what I have called a4= a+ 3d= -9 and t15= a+ 14d= -31. Solve those two equations for a and d and then tn= a+ (n-1)d.
 
  • #3


1. To find x, we can use the formula for the nth term of an arithmetic sequence, which is an = a1 + (n-1)d, where a1 is the first term and d is the common difference. In this case, we have a1 = x+3, a2 = 3x-1, and a3 = 7x-2. Plugging these values into the formula, we get:

a1 = x+3
a2 = (x+3) + d = 3x-1
a3 = (x+3) + 2d = 7x-2

Solving the first two equations for d and x, we get d = 2 and x = 5. Therefore, the terms are 8, 9, and 17 in an arithmetic sequence.

2. To find the sum of the first twenty-eight terms, we can use the sum formula for an arithmetic series, which is Sn = n/2(a1 + an), where n is the number of terms, a1 is the first term, and an is the last term. We are given the sum of the first twelve and first twenty-four terms, so we can set up the following equations:

12/2(a1 + a12) = -72
24/2(a1 + a24) = 144

Solving for a1 and a24, we get a1 = -9 and a24 = 15. Then, plugging these values into the sum formula, we get:

28/2(-9 + a28) = -72
a28 = -27

Therefore, the sum of the first twenty-eight terms is -27.

3. To find a, d, and tn, we can use the same formula for the nth term of an arithmetic sequence. We are given two terms, t4 = -9 and t15 = -31, so we can set up the following equations:

t4 = a + 3d = -9
t15 = a + 14d = -31

Solving for a and d, we get a = -3 and d = -2. Then, using the formula for tn, we can find the 4th and 15th terms:

t4 = -3 + 3(-2) = -9
t15 = -3 + 14(-2) = -31
 

1. What is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This difference is called the common difference. For example, in the sequence 2, 5, 8, 11, 14, the common difference is 3.

2. How do you find the common difference of an arithmetic sequence?

To find the common difference of an arithmetic sequence, subtract any term from the following term in the sequence. The result will be the common difference. For example, in the sequence 2, 5, 8, 11, 14, the common difference is 3 since 5 - 2 = 8 - 5 = 11 - 8 = 14 - 11 = 3.

3. What is an arithmetic series?

An arithmetic series is the sum of an arithmetic sequence. It is often denoted by Sn, where n is the number of terms in the sequence. For example, in the sequence 2, 5, 8, 11, 14, the arithmetic series with 5 terms would be 2 + 5 + 8 + 11 + 14 = 40.

4. How do you find the sum of an arithmetic series?

To find the sum of an arithmetic series, you can use the formula Sn = (n/2)(a1 + an), where n is the number of terms, a1 is the first term, and an is the last term. Alternatively, you can use the formula Sn = (n/2)(2a1 + (n-1)d), where d is the common difference.

5. How is an arithmetic sequence different from a geometric sequence?

An arithmetic sequence has a constant difference between consecutive terms, while a geometric sequence has a constant ratio between consecutive terms. In other words, in an arithmetic sequence, you add the same number each time to get the next term, whereas in a geometric sequence, you multiply by the same number each time to get the next term.

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