Use mathematical induction to prove that the following statement is true for every positive integer n
1 answer:
<h3>to prove
</h3><h3>
</h3><h3>8
</h3><h3>+
</h3><h3>16
</h3><h3>+
</h3><h3>24
</h3><h3>+
</h3><h3>...
</h3><h3>+
</h3><h3>8
</h3><h3>n
</h3><h3>=
</h3><h3>4
</h3><h3>n
</h3><h3>(
</h3><h3>n
</h3><h3>+
</h3><h3>1
</h3><h3>)
</h3><h3>−
</h3><h3>−
</h3><h3>−
</h3><h3>−
</h3><h3>(
</h3><h3>*
</h3><h3>)
</h3><h3>
</h3><h3> let </h3><h3>T
</h3><h3>n
</h3><h3>=
</h3><h3>4
</h3><h3>n
</h3><h3>(
</h3><h3>n
</h3><h3>+
</h3><h3>1
</h3><h3>)
</h3><h3>
</h3><h3>(1) verify for </h3><h3>n
</h3><h3>=
</h3><h3>1
</h3><h3>
</h3><h3>L
</h3><h3>H
</h3><h3>S
</h3><h3>=
</h3><h3>8
</h3><h3>
</h3><h3>R
</h3><h3>H
</h3><h3>S
</h3><h3>=
</h3><h3>4
</h3><h3>×
</h3><h3>1
</h3><h3>(
</h3><h3>1
</h3><h3>+
</h3><h3>1
</h3><h3>)
</h3><h3>=
</h3><h3>4
</h3><h3>×
</h3><h3>2
</h3><h3>=
</h3><h3>8
</h3><h3>
</h3><h3>∴
</h3><h3>true for </h3><h3>n
</h3><h3>=
</h3><h3>1
</h3><h3>
</h3><h3># to show
</h3><h3>
</h3><h3>T
</h3><h3>k
</h3><h3>⇒
</h3><h3>T
</h3><h3>k
</h3><h3>+
</h3><h3>1
</h3><h3>
</h3><h3>assume true for </h3><h3>T
</h3><h3>k
</h3><h3>=
</h3><h3>4
</h3><h3>k
</h3><h3>(
</h3><h3>k
</h3><h3>+
</h3><h3>1
</h3><h3>)
</h3><h3>
</h3><h3>need to show
</h3><h3>
</h3><h3>T
</h3><h3>k
</h3><h3>+
</h3><h3>1
</h3><h3>=
</h3><h3>4
</h3><h3>(
</h3><h3>k
</h3><h3>+
</h3><h3>1
</h3><h3>)
</h3><h3>(
</h3><h3>k
</h3><h3>+
</h3><h3>2
</h3><h3>)
</h3><h3>
</h3><h3>add next term to to both sides of </h3><h3>(
</h3><h3>*
</h3><h3>)
</h3><h3>
</h3><h3>8
</h3><h3>+
</h3><h3>16
</h3><h3>+
</h3><h3>24
</h3><h3>+
</h3><h3>...
</h3><h3>+
</h3><h3>8
</h3><h3>k
</h3><h3>+
</h3><h3>8
</h3><h3>(
</h3><h3>k
</h3><h3>+
</h3><h3>1
</h3><h3>)
</h3><h3>=
</h3><h3>4
</h3><h3>k
</h3><h3>(
</h3><h3>k
</h3><h3>+
</h3><h3>1
</h3><h3>)
</h3><h3>+
</h3><h3>8
</h3><h3>(
</h3><h3>k
</h3><h3>+
</h3><h3>1
</h3><h3>)
</h3><h3>
</h3><h3>∴
</h3><h3>T
</h3><h3>k
</h3><h3>+
</h3><h3>1
</h3><h3>=
</h3><h3>4
</h3><h3>k
</h3><h3>(
</h3><h3>k
</h3><h3>+
</h3><h3>1
</h3><h3>)
</h3><h3>+
</h3><h3>8
</h3><h3>(
</h3><h3>k
</h3><h3>+
</h3><h3>1
</h3><h3>)
</h3><h3>
</h3><h3>=
</h3><h3>4
</h3><h3>(
</h3><h3>k
</h3><h3>+
</h3><h3>1
</h3><h3>)
</h3><h3>[
</h3><h3>k
</h3><h3>+
</h3><h3>2
</h3><h3>]
</h3><h3>=
</h3><h3>T
</h3><h3>k
</h3><h3>+
</h3><h3>1
</h3><h3>
</h3><h3>i
</h3><h3>.
</h3><h3>e
</h3><h3>.
</h3><h3>T
</h3><h3>k
</h3><h3>⇒
</h3><h3>T
</h3><h3>k
</h3><h3>+
</h3><h3>1
</h3><h3> as required
</h3><h3>
</h3><h3>#(3) conclusion
</h3><h3>
</h3><h3>statement true for </h3><h3>T
</h3><h3>1
</h3><h3>
</h3><h3>∵
</h3><h3>T
</h3><h3>k
</h3><h3>⇒
</h3><h3>T
</h3><h3>k
</h3><h3>+
</h3><h3>1
</h3><h3>
</h3><h3>T
</h3><h3>1
</h3><h3>⇒
</h3><h3>T
</h3><h3>2
</h3><h3>⇒
</h3><h3>T
</h3><h3>3
</h3><h3>⇒
</h3><h3>...
</h3><h3>∀
</h3><h3>n
</h3><h3>∈
</h3><h3>N</h3>
You might be interested in
Answer:
Do you have a graph?
Step-by-step explanation:
Answer:
Step-by-step explanation:
63
Answer:
you can go on something called tiger algerbra
Step-by-step explanation:
p/5 + 6 = 10
=> p/5 = 10-6
=> p/5 = 4
=> p = 5×4
=> p = 20
C. 20
Answer:
$0.84
Step-by-step explanation:
1. find total cost of both the pen and paper
2. find the amount Sally spent
3. find cost of ruler
