SME
Discrete Mathematics · Problem type
Repeated root
Use this for “Solve aₙ = 4aₙ₋₁ − 4aₙ₋₂ with a₀ = 1, a₁ = 4.”
Chart
Vertex (2, 0), axis of symmetry r = 2
This is an example. Type your own values to start a new problem.
aₙ
=
c₁
aₙ₋₁
+
(
c₂
)
aₙ₋₂
a₀
=
a₀
,
a₁
=
a₁
a
=
aₙ
aₙ comes from the closed form in the Formulas: the repeated root r = c₁ ÷ 2, then A and B fitted to a₀ and a₁.
r
Repeated root
calculated
A
Constant A
worked out (not typed)
B
Constant B
worked out (not typed)
Formulas
r
=
c₁
2
2
=
4
2
c₂
=
−
r
2
−4
=
−
2
2
A
=
a₀
1 = 1
B
=
a₁
−
r
×
a₀
r
1
=
4 − 2 × 1
2
aₙ
=
(
A
+
B
×
n
)
×
r
n
32
=
(1
+
1
×
3)
×
2
3
Assumptions
•
aₙ = c₁aₙ₋₁ + c₂aₙ₋₂ with c₁² + 4c₂ = 0: then r² = c₁r + c₂ has one repeated root r = c₁ ÷ 2.
•
So c₂ = −r²: the page works c₂ out from c₁.
•
Both rⁿ and nrⁿ solve the recurrence, so every solution is aₙ = (A + Bn)rⁿ.
Step-by-step
WE KNOW
c₁ = 4, a₀ = 1, a₁ = 4, n = 3
FIND
repeated root (r), second coefficient (c₂), constant A (A), constant B (B), term n (aₙ)
Step 1 · Find repeated root (r)
Use r = c₁ ÷ 2
The
roots
of
r
2
−
c₁
r
−
c₂
=
0
add
to
c₁
;
here
both
are
r
,
so
2
r
=
c₁
.
r
=
4
2
r
=
2
Step 2 · Find second coefficient (c₂)
Use c₂ = −r²
The
roots
multiply
to
−
c₂
;
here
r
×
r
=
−
c₂
.
c₂
=
−
r
2
c₂
=
−
2
2
c₂
=
−4
Step 3 · Find constant A (A)
Use A = a₀
Put
n
=
0
in
aₙ
=
(
A
+
Bn)rⁿ:
r
0
=
1
and
Bn
=
0,
so
a₀
=
A
.
A
=
1
Step 4 · Find constant B (B)
Use B = (a₁ − r × a₀) ÷ r
Put
n
=
1:
a₁
=
(
A
+
B
)
r
with
A
=
a₀
,
so
Br
=
a₁
−
ra₀.
B
=
4 − 2 × 1
2
B
=
2
2
B
=
1
Step 5 · Find term n (aₙ)
Use aₙ = (A + B × n) × rⁿ
Put
n
into
the
closed
form:
no
need
to
work
out
every
term
before
it.
aₙ
=
(1
+
1
×
3)
×
2
3
aₙ
=
4
×
2
3
aₙ = 4 × 8
aₙ = 32
Check
Put the numbers back in. Both sides should match.
2
=
4
2
✓
−4
=
−
2
2
✓
1 = 1 ✓
1
=
4 − 2 × 1
2
✓
32
=
(1
+
1
×
3)
×
2
3
✓
Topic check
See what you know: 5 questions on this topic
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