Example 1 — summing a geometric series
Find the sum of the first 6 terms of
- 1Identify , , .Check the ratio: and ✓
- 2This form avoids two negatives when .
- 3.
- 4
- 5Check: ✓Worth doing when the term count is small.
Series
Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 2 — combining standard results
Find in terms of .
- 1Split the sum: Constants come outside; sums separate.
- 2Standard result.
- 3Adding 1 to itself times — not 1.
- 4The 2 and the denominator cancel.
- 5Check at : and ✓
Example 3 — the method of differences
Show that .
- 1Split into partial fractions: This split is the whole method.
- 2Write out the first few and the last: Three from the start and one from the end shows the pattern.
- 3Every interior term appears twice with opposite signs.The sum telescopes.
- 4What survives is The first term of the first bracket and the last of the last.
- 5Check at : ✓