Algebraic Expression for the Nth Term! So hard!

a. We see that 4=2+2, 6=4+2, 10=8+2, 18=16+2, etc. This means if the Nth term in this sequence is \(a_n\), then \(\boxed{a_n=2^n+2}\)

 

b. We see that 2=1x2, 6=3x2, 12=6x2, and 20=10x2.

 

Now if you are familiar with numbers, you may recognize immediately that 1 is the sum of the first positive integer, 3 is the sum of the first 2 positive integers, 6 is the sum of the first 3, and 10 is the sum of the first four, so the answer is the formula for this multiplied by 2, which is \(\boxed{b_n=n(n+1)}\)

 

If you are not familiar with numbers, thats ok! We can look at the sequence 1,3,6,10, and see that 3-1=2, 6-3=3, and 10-6=4. This makes us think, the next one will be 15, then 21, etc. We now know that this sequence is twice the sum of the first n positive integers, and we can find the formula as follows:

Pair up 1 and n, 2 and (n-1), 3 and (n-2), etc. You will get \(\frac{n}{2}\) pairs if n is even, and \(\frac{n-1}{2}\) pairs if n is odd, with \(\frac{n+1}{2}\) being left out. Either way, the sum is then \(\frac{n(n+1)}{2}\), and so 2 times that is \(\boxed{n(n+1)}\)

 

Hope this helped

If it's wrong please tell me how so I can learn :)

Hello friend, 

thank you for your amazing explaination. 

I think every thing is valid! And no room for improvement!