# How Do You Calculate the Sum of Specific Mathematical Progressions?

In summary, the conversation discusses finding the sum of a given progression and using the formula for the kth term to solve it. The formula for the kth term is given by 2^{k} +3 + 5(k-1). The conversation also discusses the possibility of treating the 2^{k} and 3 + 5(k-1) terms separately and how to find the sum up to any kth term. The conversation ends with a note about finding the difference between certain terms in terms of k.
(1) Let's say you have the progression $5 + 12 + 21 + ... + 1048675$ and you want to find the sum of 20 terms. I know that the kth term is given by $2^{k} +3 + 5(k-1)$. So would I treat the $2^{k}$ terms separately from the $3 + 5(k-1)$ terms? Would it be $2 + 4 + 8 + 16 +... + 2^{n}$ and $3 + 8 + 13 + ... + (3+5(k-1))$. Would the total sum be $\frac{2 - 2(2)^{n}}{-1} + \frac{n}{2}(3+ (3+5(k-1))$?

$$2097248 + 1010$$

(2) $$3 + 10 + 25 + ... + 39394$$ and you want to find sum of first 10 terms. I know that the kth term is $2 \times 3^{k-1} + 1 + 3(k-1)$ Would i do the same thing and treat the $3^{k-1}$ and $1 + 3(k-1)$ separately?

Thanks

Last edited:
I have only looked at (1) but here is what I have done:

a1 = 5
a2 = 12
a3 = 21

You have said that: ak = 2k + 5k - 2

So: ak + 1 = 2(k+1) + 5(k+1) - 2

This means that: ak+1 + ak = 2(k+1) + 5(k+1) - 2 + 2k + 5k - 2 = 2k + 1 + 2k + 10k + 1

and: This means that: ak+2 + ak+1 + ak = 2(k+2) + 5(k+2) - 2 + 2(k+1) + 5(k+1) - 2 + 2k + 5k - 2
= 2(k+2) + 2(k+1) + 2k + 15k + 9

I have found that the sum of the terms 2(k+n) for n = 0, 1, 2 etc. is 2(k+1) - 2.

The term for the sum of the 5k, 10k, 15k terms is equal to 5k2

However I was unable to work out the last term. All you need to do, from what I have done, is work out the way to express the difference between -2, 1 and 9 in terms of k when k is 1, 2 and 3 respectively. Then you can put all three terms together and you can find the sum up to any kth term.

for reaching out for help with math progressions! To find the sum of the first 20 terms in the first progression, you can use the formula for the sum of a finite geometric series: S_n = a(1-r^n)/(1-r), where a is the first term and r is the common ratio. In this case, a=5 and r=2. So the sum would be S_20 = 5(1-2^20)/(1-2) = 5(1-1048576)/(-1) = 5(1048575) = 5242875.

For the second progression, you can use the same formula to find the sum of the first 10 terms. Here, a=3 and r=3. So the sum would be S_10 = 3(1-3^10)/(1-3) = 3(1-59049)/(-2) = 3(59048)/(-2) = -88572.

## 1. What are number progressions?

Number progressions, also known as sequences, are a set of numbers that follow a specific pattern or rule. Each number in the sequence is called a term, and the pattern can be described using a formula or rule.

## 2. What are the different types of math progressions?

There are four main types of math progressions: arithmetic, geometric, harmonic, and Fibonacci. Arithmetic progressions have a constant difference between each term, geometric progressions have a constant ratio between each term, harmonic progressions have a constant difference between the reciprocals of each term, and Fibonacci progressions have each term equal to the sum of the two previous terms.

## 3. How do I find the next term in a progression?

To find the next term in a progression, you need to identify the pattern or rule that the sequence follows. Once you have the pattern, you can use it to find the next term by plugging in the previous term or terms into the formula or rule.

## 4. What is the importance of math progressions?

Math progressions are important because they can be used to model real-world situations and make predictions. They are also essential in higher-level math and can help with problem-solving and critical thinking skills.

## 5. How can I improve my understanding and skills in math progressions?

To improve your understanding and skills in math progressions, it is important to practice solving different types of progressions. You can also study and learn about the different patterns and rules that each type of progression follows. Additionally, seeking help from a tutor or teacher can also be beneficial.

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