to prove
8
+
16
+
24
+
...
+
8
n
=
4
n
(
n
+
1
)
−
−
−
−
(
*
)
let
T
n
=
4
n
(
n
+
1
)
(1) verify for
n
=
1
L
H
S
=
8
R
H
S
=
4
×
1
(
1
+
1
)
=
4
×
2
=
8
∴
true for
n
=
1
# to show
T
k
⇒
T
k
+
1
assume true for
T
k
=
4
k
(
k
+
1
)
need to show
T
k
+
1
=
4
(
k
+
1
)
(
k
+
2
)
add next term to to both sides of
(
*
)
8
+
16
+
24
+
...
+
8
k
+
8
(
k
+
1
)
=
4
k
(
k
+
1
)
+
8
(
k
+
1
)
∴
T
k
+
1
=
4
k
(
k
+
1
)
+
8
(
k
+
1
)
=
4
(
k
+
1
)
[
k
+
2
]
=
T
k
+
1
i
.
e
.
T
k
⇒
T
k
+
1
as required
#(3) conclusion
statement true for
T
1
∵
T
k
⇒
T
k
+
1
T
1
⇒
T
2
⇒
T
3
⇒
...
∀
n
∈
N
