1. Definition
For an "arithmetic × geometric" term, multiply by q and subtract the shifted series to reduce to a geometric sum. If q=1, S_n=1+…+n=n(n+1)/2.
3. Properties & Laws
- Σ k·q^(k−1);
- Multiply, shift, subtract;
- For q≠1, the closed form is (1−(n+1)q^n+nq^(n+1))/(1−q)²; for q=1, S_n=1+…+n=n(n+1)/2.
- q=2,n=5 gives 129.
4. Steps
- Enter q and n (an integer from 1 to 100,000);
- Click "Sum";
- Or click "Load Example".
5. Worked Examples
Example: Σ k·2^(k−1), n=5 gives 129.
6. Common Pitfalls
The closed form assumes q≠1; for q=1, use 1+…+n to get S_n=n(n+1)/2.
Align terms and handle the first and last terms;
The numerical sum can be checked against the closed form.