# How to Solve Arithmetic Sequence Problems

• Purpleshinyrock
In summary, the conversation discusses an arithmetic progression where the sum of the last n terms is equal to three times the sum of the first n terms. The goal is to determine the sum of the first 10 terms as a function of the difference d. After applying the formula for an arithmetic progression, the equation Sn=3sn is used to find the desired result. However, there is an error in the equation as the value of r is not specified. The conversation ends with a request for help in finding the correct solution.
Purpleshinyrock
Summary:: Sequences, Progressions

Hello. I have been Given the following exercise, Let (a1, a2, ... an, ..., a2n) be an arithmetic progression such that the sum of the last n terms is equal to three times the sum of the first n terms. Determine the sum of the first 10 terms as a function of the ratio d.

Solution is S10=50d

I know that an=a1+nd-d, and an+1=a1+nd
a2n=a1+(2n-1)d=a1+2nd-d

sn=(a1+an)(n/2)=(2a1+nd-d)(n/2)
Sn=(an+1+a2n)(n/2)=(2a1+3nd-r)(n/2)
But When I try to do Sn=3sn(the sum of the last n terms is equal to three times the sum of the first n terms) I am not getting the desired result.

[Moderator's note: Moved from a technical forum and thus no template.]

Last edited by a moderator:
A ratio is the result of a division, e.g. a/b. I assume you mean "difference" d.
Purpleshinyrock said:
Sn=(an+1+a2n)(n/2)=(2a1+3nd-r)(n/2)
What is r?
Purpleshinyrock said:
But When I try to do Sn=3sn(the sum of the last n terms is equal to three times the sum of the first n terms) I am not getting the desired result.
The approach is good so far (after fixing the r). Please show your following work, otherwise it's impossible to tell what went wrong.

## What is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers where the difference between each consecutive term is constant. This constant difference is called the common difference, and it is what makes an arithmetic sequence different from other types of sequences.

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

To find the common difference of an arithmetic sequence, you subtract any term from the term that comes after it. The result will be the common difference. For example, in the sequence 2, 5, 8, 11, the common difference is 3 because 5 - 2 = 3, 8 - 5 = 3, and so on.

## What is the formula for finding the nth term of an arithmetic sequence?

The formula for finding the nth term of an arithmetic sequence is an = a1 + (n-1)d, where an is the nth term, a1 is the first term, and d is the common difference. This formula can be used to find any term in an arithmetic sequence, given the first term and the common difference.

## How do you determine if a sequence is arithmetic?

To determine if a sequence is arithmetic, you can check if the difference between each consecutive term is the same. If it is, then the sequence is arithmetic. Another way is to use the formula for finding the nth term and see if it applies to all the terms in the sequence.

## What are some real-life applications of arithmetic sequences?

Arithmetic sequences can be used to model real-life situations that involve a constant rate of change. For example, calculating the amount of money earned by an employee who receives a fixed salary increase every year, or predicting the growth of a population that increases by a fixed number each year. Arithmetic sequences are also used in financial planning and budgeting, as well as in calculating loan payments and interest rates.

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