Advanced functions
Arithmetic and geometric sequences
ALEKS placement
Arithmetic adds a constant difference; geometric multiplies by a constant ratio.
What this covers
an=a1+(n−1)d,an=a1rn−1- Test consecutive terms: a constant difference is arithmetic, a constant ratio is geometric.
- Substitute into the matching explicit formula.
- Arithmetic sums use n/2 · (a₁ + aₙ).
- Geometric sums use a₁(1 - rⁿ)/(1 - r).
Worked example
Worked example
Find the 12th term of 5, 8, 11, 14, …
- d=3Each term adds 3.
- a12=5+(12−1)(3)Substitute into the arithmetic formula.
- a12=38Eleven gaps of 3 is 33, plus the first term 5.
Another worked example
An arithmetic sequence starts at 4 and adds 3. Find a₈ and the sum of its first eight terms.
- a8=4+(8−1)3=25Eight terms use seven changes from the first value.
- S8=28(4+25)=116The finite sum is the count times the average of the first and last terms.
A common mistake
A mistake Lemma catches
From d=3
a12=5+12(3)a12=5+(12−1)(3)
“From the 1st term to the 12th there are 12 − 1 = 11 steps of 3, because the first term takes no steps: 5 + 11 · 3 = 38. Using 12 steps adds one d too many: 5 + 12 · 3 = 41, which is the 13th term.”
Try one
Sample problem
Find a_9.
a1=3,d=2Practice sequences free
No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.
Where this fits
Before this