I've been studying arithmetic sequences and am finding that I can do the formulas, but can't truly understand until I can do a long-form version of the formula.
Let's take the below:
$a$5 = 2+2(5-1)
I can do 2+2+2+2+2 to get to 10 or following the sequence step by step I can do 2,4,6,8,10. But how would I calculate these answers on something like a basic calculator?
Edit: In case unclear, I'm asking for the non-formulaic version to calculate what, given the above, would write as $a$n = $a$1 + d (n – 1).
Edit 2: Okay, say I give you any arithmetic sequence formula such as shown above, and I hand you a calculator as basic as the one shown in this image. How do you solve it? You don't just do $a$1 + $a$2 + $a$3 … ad infinitum. How do you solve for it? All the more advanced calculators and services online provide none of the step-by-step process, so you get the final answer without understanding a thing, and there has to be a more effective way than just adding or subtracting to the nth term.
Best Answer
So the formula $$a_n = a_1 + d(n-1)$$ is the explicit formula, or closed form. You don't need to calculate $a_4$ to calculate $a_5$. For example if we have the same formula as you have in your post, $a_n = 2+ 2(n-1)$ then if you want to calculate $a_{10}$ you just plug in 10 to the equation and compute it using a calculator if you like $$a_{10}= 2 + 2(10-1)$$