Step 1 of 10
Start
Read the recursion as one repeatable move.
Decode the rule
a1 = 4 and an = an−1 + 3
Find a5 when the sequence starts at 4 and adds 3 each time.
Live topic example: f(1)=2, f(n)=f(n−1)+7. Find f(10).
Read rule
›
Build terms
›
Count links
›
Land term
Recursive rule · previous term + common difference
A recursive rule is a term-to-term machine
It tells you where to start and what identical move creates the next term.
TERM TRANSIT · RULE DECODERADD 3
an−1the previous station
+ 3same move
anthe next station
To make the next term, take the term immediately before it and add the common difference 3.
a1 = 4start station
an−1term before
d = +3equal jump
Recursive means “use the previous term.”next = previous + 3
Build the route, then count its links
Term numbers label stations. The common difference labels the equal jumps between them.
INDEXED TERM ROUTEa1=4 · d=3
a14
+3
a27
+3
a310
+3
a413
+3
a516
a10 jumps
a21 jump
a32 jumps
a43 jumps
a54 jumps
The first term is already standing at station 1. To reach station 5, cross only 4 links: 5 − 1.
ROUTE LEDGER
starta1 = 4
each linkd = +3
to a55 − 1 = 4 links
arrivala5 = 16
Count links first: n − 1
an =a1+(n−1)d
start valuerepeat dnumber of links
4 equal links × 3 per link
a5 =4+(5−1)×3=16
Your turn: ride from term 1 to term 10
Use the live topic example. Record the start, jump, number of links, total jump, and arrival value.
f(10) = start + links × d
Fill all five route entries. Remember: term 1 is the starting station.
✎
Solve on your whiteboard
Write f(10) = f(1) + (10−1)d, substitute the values, and find the tenth term. Submit when your route is complete.
Work submitted. The solution route is unlocked.
Nine equal links land at 65
The tenth station needs nine moves because the first term is already given.
TERM 1 TO TERM 10d = +7
f(1)2
+7, +7
···
9 equal links
f(10)65
10 stations have 9 gaps between station 1 and station 10. That is why the formula uses 10 − 1.
ARRIVAL CALCULATION
readf(1)=2, d=7
10 − 1 = 9 links
links9 × 7 = 63
Add total jump to start
arrival2 + 63 = 65
not 10 links2 + 10×7 = 72 ✕
Recursive-term route complete
You can generate nearby terms or jump straight to a distant term without losing the meaning of the recursion.
n − 1
Reliable rule: start at a1, count n−1 equal links, multiply by the common difference d, then add that total jump to the start.
Read startRead dCount n−1 linksLand on an