Determine the sum of the first 20 terms of the arithmetic sequence in which ...

f) the 7th term is 43 and the 13th term is 109

I know you can make two equations :

43 = a + 6d and 109 = a + 12 d

And subtract them to get d = 11

But I know there's another way to get d, that you can just quickly do on your calculator. But I can't remember how to do it! It had something to do with dividing the number of terms or something...? I could probably figure it out after some trial and error but I just want to make sure I have the proper method down, so if someone could explain a clear and concise method to me, I would really appreciate it! I know it's really simple... I'm just so burned out right now. Too much studying, too much stress!

There are 6 terms from term7 to term13

and the difference in their values is 109-43 = 66
so the common difference is 66/6 = 11

but, ... , notice that you are actually doing the same calculations you would do if you solved the two equations, so nothing gained.

I think this way is a lot faster and easier to do by just quickly typing it all into the calculator, so thank you!

Ah, burnout and stress, the classic combo! Don't worry, I'm here to lighten the mood and help you out. Let's tackle this arithmetic sequence problem together with a touch of humor!

So, you're correct with those two equations. Good job on that! Now, to find the common difference, d, there's actually a quicker way than trial and error. It's like finding a shortcut in a maze, but without any corny monsters chasing you!

Here's the clear and concise method you're looking for: You take the difference between the seventh term (43) and the first term (a), and then divide it by the number of terms before the seventh term.

In this case, the number of terms before the seventh term is 6 (since the seventh term is the 7th term, duh!). So, we have (43 - a) / 6 = d. Simple, right? No need to divide your brain cells into a frenzy!

Solving this equation leads to the amazing discovery that d equals 11. Ta-da! You're back on track without any trial and error headaches.

Now that you have d figured out, calculating the sum of the first 20 terms becomes as easy as pie... or as easy as chuckling at a good clown joke! Just plug the values into the formula for the sum of an arithmetic sequence: S = (n/2)(2a + (n - 1)d).

For this problem, the first term, a, is something we don't know yet, but that's not a problem! We have the seventh term (43) and the common difference (11), so we can use the equation 43 = a + 6d to solve for a. After substituting the values, you'll find that a equals 37.

Now, with a = 37 and d = 11, you can calculate the sum, S, with the equation S = (20/2)(2(37) + (20 - 1)(11)). Crunch those numbers, and you'll have your answer!

Remember, when the stress gets to you, just bring in a little humor and clown around with the problem. It'll make the journey much more enjoyable! Good luck, my friend!

I understand that you're feeling burned out, but I'm here to help you. Don't worry, I'll guide you through the steps to find the common difference (d) in a clearer and quicker way.

To find the common difference, we can use the formula:

d = (a_n - a_1) / (n - 1)

Where:
- d is the common difference
- a_n is the nth term
- a_1 is the first term
- n is the number of terms

In this case, we have the 7th term (a_7 = 43) and the 13th term (a_13 = 109). We need to find d.

Let's calculate:

d = (a_13 - a_7) / (13 - 7)
= (109 - 43) / 6
= 66 / 6
= 11

So the common difference (d) is 11. Your initial calculation was correct!

Now, if you want to find the sum of the first 20 terms, you can use the formula for the sum of an arithmetic sequence:

S_n = (n/2) * (a_1 + a_n)

Let's calculate:

S_20 = (20/2) * (a_1 + a_20)
= 10 * (a_1 + (a_1 + (n-1)d))
= 10 * (2a_1 + (n-1)d)
= 10 * (2a_1 + (20 - 1) * 11)
= 10 * (2a_1 + 19 * 11)

Now, if you have the value of the first term (a_1), you can substitute it into the equation to find the sum (S_20).

I understand that you're looking for a simple method to find the common difference (d) of an arithmetic sequence. There is indeed a formula that can help you do this quickly, and it involves using the given terms of the sequence.

To find the common difference (d) of an arithmetic sequence, you can use the formula:

d = (aᵢ₊ₙ - aᵢ) / n

Where:
- d is the common difference
- aᵢ₊ₙ is the nth term of the sequence
- aᵢ is the initial term of the sequence
- n is the number of terms between the two given terms (inclusive)

In this case, we are given the 7th term (a₇ = 43) and the 13th term (a₁₃ = 109). Now, let's apply this formula to find the common difference:

d = (a₁₃ - a₇) / (13 - 7)

d = (109 - 43) / 6

d = 11

So, the common difference (d) for this arithmetic sequence is 11, as you correctly mentioned.

It's great that you remembered the strategy of using two equations to find the common difference as well! Keep in mind that both methods will yield the same result.

Now, if you want to find the sum of the first 20 terms, you can use the formula:

Sn = (n/2)(2a + (n - 1)d)

Where:
- Sn is the sum of the first n terms
- n is the number of terms
- a is the first term
- d is the common difference

Substituting the given values, we have:

S20 = (20/2)(2a + (20 - 1)d)

S20 = 10(2a + 19d)

S20 = 10(2a + 209) [since d = 11]

Now, you would need to have the value of the initial term (a) to complete the calculation of the sum. If you have that value, you can proceed with substituting it into the equation to get the final answer for the sum of the first 20 terms.