SME
Grade 12 Math · Precalculus
Mathematical induction
Use this for “Prove 1 + 2 + … + n = n(n + 1)/2 by induction,” checking the step from n to n + 1.
Chart
Terms: 1, 2, 3, 4 (add d = 1 each time).
a₄ = 1 + 3 × 1 = 4
S₄
=
4
×
1 + 4
2
=
10
This is an example. Type your own values to start a new problem.
1
+
2
+
…
+
n
=
(
+
1)
2
=
S
a
Next term
worked out (not typed)
T
S plus the next term
worked out (not typed)
F
The formula at n + 1
worked out (not typed)
Formulas
S
=
n
×
n
+ 1
2
10
=
4
×
4 + 1
2
a
=
n
+
1
5 = 4 + 1
T
=
S
+
a
15 = 10 + 5
F
=
(
n
+
1)
×
n
+ 2
2
15
=
(4
+
1)
×
4 + 2
2
Assumptions
•
Base case: check the formula at n = 1.
•
Step: assume it holds at n = k; adding term k + 1 must give the formula at k + 1.
•
Here: k(k + 1)/2 + 2(k + 1)/2 = (k + 1)(k + 2)/2. Numbers check one case; the algebra proves every case.
Step-by-step
WE KNOW
n = 4
FIND
sum of the first n terms (S), next term (a), S plus the next term (T), the formula at n + 1 (F)
Step 1 · Find sum of the first n terms (S)
Use S = n × (n + 1) ÷ 2
The
formula
at
n
:
1
+
2
+
…
+
n
=
n
(
n
+
1)/2.
S
=
4
×
4 + 1
2
S
=
4
×
5
2
S
=
20
2
S
=
10
Step 2 · Find next term (a)
Use a = n + 1
The
next
term,
number
n
+
1:
the
number
n
+
1
itself.
a
=
4
+
1
a
=
5
Step 3 · Find S plus the next term (T)
Use T = S + a
If
the
formula
holds
at
n
,
the
sum
through
term
n
+
1
is
S
plus
the
next
term.
T
=
10
+
5
T
=
15
Step 4 · Find the formula at n + 1 (F)
Use F = (n + 1) × (n + 2) ÷ 2
The
formula
with
n
+
1
in
place
of
n
.
F
=
(4
+
1)
×
4 + 2
2
F
=
5
×
6
2
F
=
30
2
F
=
15
→
T
=
F
=
15:
adding
the
next
term
gives
the
formula
at
n
+
1.
With
k
for
n
:
k(k
+
1)/2
+
2(k
+
1)/2
=
(k
+
1)(k
+
2)/2
Check
Put the numbers back in. Both sides should match.
10
=
4
×
4 + 1
2
✓
5 = 4 + 1 ✓
15 = 10 + 5 ✓
15
=
(4
+
1)
×
4 + 2
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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