I need to find the equations of the following, using Σ (and the n=1 which is at the bottom of the Σ) (and infinity is at the top of the Σ)
1) 5/1 + 8/1 + 1/1 + 14/1 + . . .
2) 6/5 + 9/8 + 14/11 + 21/14 + . . .
3) -6/3 - 4/4 - 2/5 - 0 + . . .
4) 9/1 + 36/4 +27/9 + 144/16 + . . .
5) 20/4 + 25/5 + 30/6 + 35/7 +40/8 + ...
To find the equations for these series using sigma notation, we can observe the patterns in each series. Let's break down each series step by step:
1) 5/1 + 8/1 + 1/1 + 14/1 + ...
In this series, we notice that the numerator is increasing by 3 each time, and the denominator is constant at 1. We can write the general term of this series as (3n + 2) / 1. Using sigma notation, the equation for this series becomes:
∑((3n + 2) / 1) for n = 1 to ∞
2) 6/5 + 9/8 + 14/11 + 21/14 + ...
In this series, we observe that both the numerator and denominator are increasing. The numerator increases by 3 each time, and the denominator increases by 3 starting from 5. We can write the general term of this series as (3n + 3) / (3n - 1). Using sigma notation, the equation for this series becomes:
∑((3n + 3) / (3n - 1)) for n = 1 to ∞
3) -6/3 - 4/4 - 2/5 - 0 + ...
In this series, we can see that both the numerator and denominator are decreasing. The numerator decreases by 2 each time, and the denominator increases by 1 starting from 3. We can write the general term of this series as (-2n + 6) / (n + 2). Using sigma notation, the equation for this series becomes:
∑((-2n + 6) / (n + 2)) for n = 1 to ∞
4) 9/1 + 36/4 + 27/9 + 144/16 + ...
In this series, we notice that the numerator and denominator both follow patterns. The numerator is the square of the previous term's numerator, and the denominator is always a perfect square. We can write the general term of this series as (9^n) / (4^n). Using sigma notation, the equation for this series becomes:
∑((9^n) / (4^n)) for n = 1 to ∞
5) 20/4 + 25/5 + 30/6 + 35/7 + 40/8 + ...
In this series, we can observe that the numerator increases by 5 each time, and the denominator increases by 1 starting from 4. We can write the general term of this series as (5n + 15) / (n + 3). Using sigma notation, the equation for this series becomes:
∑((5n + 15) / (n + 3)) for n = 1 to ∞
These are the equations for the given series using sigma notation.