Sum of an Arithmetic Sequence × a Geometric Sequence
A sum such as S = 1·x + 2·x² + 3·x³ + … + n·xⁿ
can be transformed into the form of a geometric series by taking the difference between S and xS.
Example ① Calculate S = 1·2 + 2·2² + 3·2³ + 4·2⁴ + 5·2⁵.
Solution:S = 2 + 8 + 24 + 64 + 160 = 258
（Alternatively, a formula can be derived by using S − 2S）
Solution:S = 2 + 8 + 24 + 64 + 160 = 258
（Alternatively, a formula can be derived by using S − 2S）
Key Point
- “Subtract xS from S” is a standard technique
Practice Problems
- S = 1·1 + 2·2 + 3·3 + 4·4 = ?
Answer and Explanation
- Answer:30
The sum of k², 4·5·9/6=30.