Each of these sequence or series formulas involves four quantities. For each formula, describe the four quantities. Then explain how you can find the forth quantity if you know the values of the other three.
a)An=a1+(n-1)d
b)Sn=n/2(a1+an)
c)An=a1r^(n-1)
d)Sn=(A1(1-r^n)/(1-r)
a) For the arithmetic sequence formula, the four quantities are:
- An: The nth term of the sequence.
- a1: The first term of the sequence.
- d: The common difference between consecutive terms.
- n: The position of the term in the sequence.
To find the fourth quantity (An) if you know the values of the other three, you can use the formula:
An = a1 + (n - 1)d.
b) For the formula of the sum of an arithmetic series, the four quantities are:
- Sn: The sum of the first n terms of the series.
- a1: The first term of the series.
- an: The nth term of the series.
- n: The number of terms in the series.
To find the fourth quantity (Sn) if you know the values of the other three, you can use the formula:
Sn = n/2 * (a1 + an).
c) In the geometric sequence formula, the four quantities are:
- An: The nth term of the sequence.
- a1: The first term of the sequence.
- r: The common ratio between consecutive terms.
- n: The position of the term in the sequence.
To find the fourth quantity (An) if you know the values of the other three, you can use the formula:
An = a1 * r^(n - 1).
d) Finally, for the formula of the sum of a geometric series, the four quantities are:
- Sn: The sum of the first n terms of the series.
- a1: The first term of the series.
- r: The common ratio between consecutive terms.
- n: The number of terms in the series.
To find the fourth quantity (Sn) if you know the values of the other three, you can use the formula:
Sn = (a1 * (1 - r^n))/(1 - r).