Jump directly to any term
The nth-term rule reaches a distant term without listing every earlier term.
Learning goals
- Explain and use a_n=a+(n-1)d.
- Find specified terms and valid term positions.
- Recover an AP from term conditions.
Concept lesson
Identify a and signed d
For 7,3,-1,..., a=7 and d=3-7=-4. Preserve the sign before using a formula.
Why n-1 differences
The first term uses zero jumps, the second one jump and the nth term n-1 jumps. Hence a_n=a+(n-1)d; substituting n=1 must return a.
Find a distant term
For 2,7,12,..., a=2,d=5. Then a15=2+14(5)=72.
Which term is 47?
For 2,7,12,... solve 2+(n-1)5=47. This gives n=10, a positive integer, so 47 is the tenth term.
Recover the general term
If a=1 and a14=53, then 1+13d=53, so d=4. Therefore a_n=1+(n-1)4=4n-3.
Check n=1
Test each proposed nth-term formula at n=1 and n=2. Reject any formula that fails to reproduce a and a+d.
Is the index valid?
Solve for n when testing three candidate values. Classify each result as valid positive integer, fractional or non-positive, and state whether the value is a term.
Count divisible numbers
Identify the first and last multiples in the interval as AP terms, then solve l=a+(n-1)d for the count. Respect whether endpoints are included.
Two-mark nth-term structure
State a and d, write a_n=a+(n-1)d, substitute, solve, and validate n when it represents a position.
Exit ticket
For 100,96,92,... find a20; determine whether 14 is a term; and derive a_n in simplified form.