A term is not a total
a_n names one term; S_n totals the first n terms. Keeping these meanings separate prevents most sum errors.
Learning goals
- Select and apply the appropriate AP sum formula.
- Find missing AP quantities from sum conditions.
- Derive terms and tail sums from partial sums.
Concept lesson
Find the last term when needed
If l is not directly given, calculate l=a_n=a+(n-1)d before using S_n=n(a+l)/2.
Choose the sum formula from available data
Use S_n=n/2[2a+(n-1)d] when a,d,n are known. Use S_n=n/2(a+l) when a,l,n are known. They are equivalent because l=a+(n-1)d.
Direct sum using a and d
For 5,9,13,..., a=5,d=4,n=20. S20=10[10+19(4)]=860.
Find a from S20 and a20
Given S20=820 and l=a20=79, use 820=20(a+79)/2. Thus a+79=82 and a=3.
Recover a term from partial sums
S_n includes a_n while S_(n-1) stops before it, so a_n=S_n-S_(n-1). For S4=20,S3=12, a4=8.
Match data to formula
For four data sets, label known a,d,l,n and select the formula requiring no unnecessary intermediate calculation.
Term versus total
Label each symbol as a single term or accumulated total, then complete a_n=S_n-S_(n-1) and verify with a short numerical AP.
Sum of the last ten terms
For a 59-term AP, the last ten terms are terms 50 through 59. Their sum is S59-S49, not S59-S50.
Sum-answer structure
State known quantities, select and write the formula, substitute visibly, and for a tail sum identify the exact preceding partial sum.
Exit ticket
Find S20 for 6,10,14,...; derive a10 from S10 and S9; and state which partial sums give the sum of terms 21 through 30.