# Can You Solve This Exercise on Arithmetic-Geometric Series?

• B
• Purpleshinyrock
In summary, the conversation is about finding the sum of a sequence using the formula (10^n - 1)/9 and the tip given about geometric series. The person asking for help is not familiar with the notation of summation and is seeking clarification on how to apply the formula to solve the given problem. They are also asked to use the formulas for sums of sums, sums of constant multiples, and sums of constants, and the summation formula for a geometric series.
Purpleshinyrock
TL;DR Summary
sequences,
Hello, I am currently self studying sequence and series and I got to a topic called arithmetic-geometric sequence, and after the theory It gives this exercise:

1) Find the sum:
S=1+11+111+1111+...+111...111, if the last (number) is a digit of n.

I was given a tip That says that
1 = (10 - 1)/9

11 = (100 - 1)/9 = (102 -1)/9

111 = (1000 - 1)/9 = (103 -1)/9

...

1111...111 = (100..000 - 1)/9 = (10n -1)/9

But I don't get how They got to this law of formation, did they apply a formula, what did they do?
Thank You.

Your sum is ##1+11+111+\ldots+11\ldots 111 =\displaystyle{\sum_{k=1}^n \left(\dfrac{10^n}{9}-\dfrac{1}{9}\right)}##.

What do you know about ##\sum_k (a_k+b_k)\, , \, \sum_k (c\cdot a_k)\, , \,\sum_k c## and geometric series?

fresh_42 said:
Your sum is ##1+11+111+\ldots+11\ldots 111 =\displaystyle{\sum_{k=1}^n \left(\dfrac{10^n}{9}-\dfrac{1}{9}\right)}##.

What do you know about ##\sum_k (a_k+b_k)\, , \, \sum_k (c\cdot a_k)\, , \,\sum_k c## and geometric series?
I do not recognize the summation, And About the geometric series I know of their general term,common ratio, sum of elements

Purpleshinyrock said:
I do not recognize the summation, And About the geometric series I know of their general term,common ratio, sum of elements
##\sum_{k=1}^n (a_k+b_k)= (a_1+b_1)+(a_2+b_2)+\ldots +(a_n+b_n)## explains the notation with ##\Sigma##.
It is a short way to write sums without dots in between.

Given that, can you get a formula for:
\begin{align*}
\sum_{k=1}^n (a_k+b_k)& = \ldots \\
\sum_{k=1}^n (c\cdot a_k)& = \ldots \\
\sum_{k=1}^n c & = \ldots
\end{align*}
If you understand how sums of sums, sums of constant multiples, and sums of constants behave, then you can apply this to your formula. Finally you will need the summation formula for a geometric series:
$$\sum_{k=1}^n q^n = \ldots$$
These are the formulas you need to solve the question given the hint.

Purpleshinyrock

## 1. What is an arithmetic-geometric series?

An arithmetic-geometric series is a series in which each term is the product of a constant difference and a constant ratio. This means that each term in the series is found by adding a fixed number to the previous term and then multiplying by a fixed number.

## 2. How is an arithmetic-geometric series different from an arithmetic series?

An arithmetic series has a constant difference between each term, while an arithmetic-geometric series has both a constant difference and a constant ratio between each term.

## 3. What is the formula for finding the sum of an arithmetic-geometric series?

The formula for finding the sum of an arithmetic-geometric series is S = a/(1-r) + (d/(1-r)^2), where S is the sum, a is the first term, r is the common ratio, and d is the common difference.

## 4. Can an arithmetic-geometric series have an infinite number of terms?

Yes, an arithmetic-geometric series can have an infinite number of terms as long as the common ratio is less than 1.

## 5. How is an arithmetic-geometric series used in real life?

Arithmetic-geometric series are used in many real-life applications, such as calculating compound interest, population growth, and the value of investments over time.

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