The formula for the nth term of the sequence of partial sums, often denoted as Sn, depends on the sequence you're summing. Here, we'll discuss the general concept and then provide specific formulas for common sequences like arithmetic and geometric sequences.
Understanding Partial Sums
A partial sum is the sum of a finite number of terms in a sequence. The nth partial sum, Sn, is the sum of the first n terms of the sequence.
General Representation
If the original sequence is given by a1, a2, a3, ..., an, then the nth partial sum, Sn, is:
Sn = a1 + a2 + a3 + ... + an
This can also be expressed using summation notation:
Sn = ∑i=1n ai
Partial Sum Formulas for Specific Sequences
The formula for Sn changes depending on the type of sequence (arithmetic, geometric, etc.).
Arithmetic Sequence
An arithmetic sequence has a constant difference (d) between consecutive terms. If the first term is a1, the nth term is an = a1 + (n-1)d. The nth partial sum is:
*Sn = (n/2) (a1 + an)**
or
*Sn = (n/2) [2a1 + (n-1)d]**
Geometric Sequence
A geometric sequence has a constant ratio (r) between consecutive terms. If the first term is a1, the nth term is an = a1 * r(n-1). The nth partial sum is:
*Sn = a1 (1 - rn) / (1 - r)** (where r ≠ 1)
If r = 1, then Sn = n * a1
Other Sequences
For sequences that don't fall into the arithmetic or geometric categories, you often need to derive a specific formula for the partial sum based on the pattern of the sequence. This might involve techniques like:
- Telescoping Series: Terms cancel out in a way that simplifies the sum.
- Mathematical Induction: Prove a formula for Sn is correct by showing it holds for n=1 and that if it holds for n=k, it also holds for n=k+1.
Example
Let's find the 5th partial sum of the arithmetic sequence 2, 4, 6, 8, 10...
Here, a1 = 2, d = 2, and n = 5.
a5 = 10
Using the formula: Sn = (n/2) * (a1 + an)
S5 = (5/2) (2 + 10) = (5/2) 12 = 30
Thus, the sum of the first 5 terms is 2 + 4 + 6 + 8 + 10 = 30.
Conclusion
The formula for the nth term of the sequence of partial sums, Sn, is dependent on the nature of the original sequence. While a general representation exists using summation notation, specific formulas are readily available for common sequence types like arithmetic and geometric sequences. For other sequences, techniques like telescoping series or mathematical induction might be required to find a formula for Sn.
