Question

A sequence $b_0,b_1,b_2,\cdots$ is defined by letting $b_0=5$ and $b_k=4+b_{k-1}$ for all natural numbers $k.$ Then which of the following is/are correct?
(Solve using mathematical induction)

• A. $b_4-b_3=4$
• B. $b_5-b_3=10$
• C. $b_n=4n+5$
• D. $b_n=5n+4$

Answer 1

Answer: (A), (C)

$b_n=4n+5$

From mathematical induction : the given sequence is true for $k=1,2,3,\cdots$
For $k=1,b_1=4+b_0=4+5$
For $k=2,b_2=4+b_1=4+(4+5)=2\cdot 4+5$
for $k=3,b=3=4+b_2=3\cdot 4+5$
As, the sequence is true for $k=1,2,3$
Let $P\left(n\right):b_n=4n+5$ is true for $k=n$
Now, for $k=n+1:$
$p(n+1)=b_{n+1}=4+b_n$
$\Rightarrow b_{n+1}=4+4n+5=4(n+1)+5$
which is also true for $k=n+1$
$\therefore b_n=4n+5$ is true for all $n\in N$