+D) (27)
11.13 +24+3.5+
+(2) solve by induction
Answers
Answer:
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Step-by-step explanation:
Answer:
Let
P(n):
2.7
n
+
3.5
n
−
5
is divisible by 24
We note thatP(n)is true when
n=
1
, since
2.7+
3.5−
5=
24
. which is divisible by
24.
Assume that P(k) is true.
i.e.
2.7
k
+
3.5
k
−
5=
24q
when
q∈
N
-------------- ( 1 )
Now, we have to prove that
P(k+
1)
is true whenever
P(k) is true.
We have
2.7
k+1
+ 3.5
k+1
− 5
⇒
2.7
k
.7
1
+
3.5
k
.5
1
−
5
⇒
7[2.7
k
+
3.5
k
−
5−
3.5
k
+
5]+
3.5
k
.5−
5
⇒
7[24q−
3.5
k
+
5]+
15.5
k
−
5
⇒
2×
24q−
21.5
k
+
35+
15.5
k
−
5
⇒
7×
24q−
6.5
k
+
30
⇒
7×
24q−
6(5
k
−
5)
⇒
7×
24q−
6(4p)
[
(5
k
−
5)
is multiple of
4
]
⇒
7×
24q−
24p
⇒
24(7p−
q)
⇒
24×
r;r=
7p−
q.
is some natural number ---------- ( 2 )
The expression on the R.H.S oof ( 1 ) is divisible by 24 . Thus
P(k+
1)
is true whenever P(k) is true.
Hence, by principle of mathematical induction , P(n)is true for all
n∈
N.
Step-by-step explanation:
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