A reliable method
For arithmetic sequences start with a₁ + (n−1)d. For a sum, pair first and last terms. For geometric sequences multiply by the ratio once for each transition.
The common trap
The nth term is not the sum of the first n terms. A deposit made at the beginning of a schedule may experience a different number of growth intervals.
Check the conclusion
Write the first three terms and check the indexing. For a finite sum, explicitly add a short example to verify the formula before applying it to larger numbers.
Contrast the cases
For the arithmetic sequence 5,8,11,…, the fourth term is 14 because three increments separate the first and fourth terms. The sum of the first four is 38, a different request. For the geometric sequence 2,6,18,…, term five is 2×3⁴=162. Replacing repeated multiplication by repeated addition would produce 14, the wrong type of sequence.
Transfer and explain
Transfer: payments 2,4,6,8,10 total 30, found by five times their average 6. Compare that sum with the final payment 10 and explain why a table’s last row does not by itself represent the accumulated total.
An arithmetic sequence begins with 9 and has common difference 4. What is its eighth term?
- From the first to the eighth term there are seven changes.
- Add seven copies of 4 to 9.
- The eighth term is 37.
Compare the other answer choices
Counts one increment too few.
Counts n increments instead of n−1.
Adds n copies of the first term, n×a, instead of advancing to the nth term with n−1 differences.
Returns the sum rather than the nth term.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GMAT exam contentSiyavula Grade 10 mathematics foundation reference