First plus last, times half the number of terms
S = (n/2)(a + l) works even when d is not known.
Only the ends are known
An AP has 8 terms. Its first term is 3 and its last term is 31. You are not told d.
Pair the first term with the last, the second with the second-last, and so on.
What will the totals of the pairs be?
What this lesson covers
The idea
If l is the last (nth) term of an AP, the sum of its n terms is S = (n/2)(a + l), useful when the first and last terms are known but d is not.
Only the ends are known
An AP has 8 terms. Its first term is 3 and its last term is 31. You are not told d.
Pair the first term with the last, the second with the second-last, and so on.
What will the totals of the pairs be?
- All the same
- Bigger and bigger as we go in
- Smaller and smaller as we go in
Fold the list in half
Tap each pair to add it. When all are done, press Hide the middle to see whether you still need d.
First and last
S = (n/2)(a + *l*), where *l* is the last term (the nth term) of the AP.
Every pair adds to a + *l*: when one number is 1 step above the first term, its partner is 1 step below the last term, so d cancels. There are n ÷ 2 pairs. This is useful when you know the first and last terms but not d.
It is the same formula as before: *l* = a + (n − 1)d, so 2a + (n − 1)d = a + *l*.
Notes
S = (n/2)(a + *l*), where *l* is the last term. Every pair (first + last, second + second-last, …) adds to a + *l*. It is useful when d is not given.
Check yourself
An AP has 10 terms. Its first term is 4 and its last term is 40. Find the sum of all its terms.
Answer: 220
S = (10/2)(4 + 40) = 5 × 44 = 220.
An AP has 5 terms. Its first term is 6 and its last term is 30. Find the sum.
Answer: 90
S = (5/2)(6 + 30) = (5/2) × 36 = 90.
When is S = (n/2)(a + *l*) more handy than S = (n/2)[2a + (n − 1)d]?
A flower bed has 23 rose plants in the first row, 21 in the second, 19 in the third, and so on, down to 5 plants in the last row. There are 10 rows. How many plants are there in all?
Answer: 140
S = (10/2)(23 + 5) = 5 × 28 = 140.
- When you know the first and last terms but not d — correct. Yes! You only need a, *l* and n.
- When you know d but not the last term. Then the other formula, with 2a + (n − 1)d, is the handy one.
- Never. They give different answers.. They always give the same answer, because *l* = a + (n − 1)d.