Who invented the nth term

The sum of an arithmetic sequence is the sum of all the terms in it. We use the first term a , the common difference d , and the total number of terms n in the AP to find its sum. Learn Practice Download. Sum of n Terms of an AP An arithmetic progression is a sequence of numbers or variables in which the difference between consecutive terms is the same.

Sum of First n Terms of an Arithmetic Progression 2. Sum of n Terms of AP Formula 3. Sum of n Terms of AP Proof 4. Sum of n Terms in an Infinite AP 5. Want to build a strong foundation in Math? Experience Cuemath and get started. Practice Questions. Explore math program.

Explore coding program. Sum of n Terms of an AP Worksheets. Looks like we nailed the conjecture. You might like to think about the following questions before you go on. But we are dividing by 2 on the Right Hand Side of the equation. Suppose that we wanted to add up all the numbers from 8 to How could we do that? Method 2: On the other hand, we could first add 1 to 7 and then 1 to 93 using our formula.

Then we could subtract the smaller from the larger. Like this:. In other words, can you generalise conjecture? This is the way that mathematics always tries to go to increase what we know about the world. What is the next way to extend what we are doing? Can we make any progress if the steps are bigger than one?

What might this formula be in words? Let them work on it for a moment. Then let them know that an 8 year old can do this in his head. That should lead to a hunt to find some patterns in the numbers 1 to that might make it easier to do the summation quickly.

But you can add the numbers from 1 to together quickly this other way too. Then try them with other examples. If you arm yourself with a calculator you might challenge them to add the numbers from 1 to anything they choose quicker than you can. Get them to do a few examples and ask them to conjecture what the pattern is. Depending how well things are going you might press on to some Arithmetic Progressions where the common difference is not 1 see section 8.

With an interested group you could even do the proof. But you should say something about Gauss and his importance on the mathematical scene. You should also find some interesting stories about him in the link I gave above. A bit of history never goes astray. In this section we complete the work that we did in Sections 2 to 6. Of course, how far you go with this problem will depend on the algebraic confidence of your staff, though you can bypass algebra completely if you think that it will go over like a lead balloon.

Anyway, try to push them a little out of their comfort zone but throw them a lifeline when they are drowning. We leave that decision to you but there ought to be enough material here for your seminar.

What I am going to try to do now is to find a formula for the sum of a string of numbers where the step up from one number to the next is always the same.

We can do this in several ways. Straight adding is one, as is dividing all of the numbers in the sum by 2 and using the formula that we know. So, forwards then backwards we have. The only problem now is how many terms are there?

Since there are 34 terms here, there have to be 34 terms in S. You might like to check this by one of the other methods but it does look a bit like the formula that we found in section 5.

Straight adding is one, as is dividing all of the numbers in the sum by 3 and using the formula that we know. You might like to check this by one of the other methods.

The most well-known story is a tale from when Gauss was still at primary school. The teacher couldn't understand how his pupil had calculated the sum so quickly in his head, but the eight year old Gauss pointed out that the problem was actually quite simple. It is remarkable that a child still in elementary school had discovered this method for summing sequences of numbers, but of course Gauss was a remarkable child. Fortunately his talents were discovered, and he was given the chance to study at university.

By his early twenties, Gauss had made discoveries that would shape the future of mathematics. While the story may not be entirely true, it is a popular tale for maths teachers to tell because it shows that Gauss had a natural insight into mathematics.

Rather than performing a great feat of mental arithmetic, Gauss had seen the structure of the problem and used it to find a short cut to a solution.