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The solution to the recurrence relation an = an-1 + 2n, with initial term a0 = 2 are _________
- a)4n+7
- b)2(1+n)
- c)3n2
- d)5*(n+1)/2
Correct answer is option 'B'. Can you explain this answer?
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The solution to the recurrence relation an= an-1+ 2n, with initial ter...
Solution to Recurrence Relation an= an-1 2n
Given recurrence relation is an= an-1 2n, with initial term a0= 2. We need to find the solution for this recurrence relation.
1. Finding the first few terms of the sequence
Let's find the first few terms of the sequence using the recurrence relation.
a0 = 2 (given)
a1 = a0 + 2^1 = 2 + 2 = 4
a2 = a1 + 2^2 = 4 + 4 = 8
a3 = a2 + 2^3 = 8 + 8 = 16
a4 = a3 + 2^4 = 16 + 16 = 32
2. Observing the pattern
Looking at the first few terms, we can observe that an = 2(1+n). Let's prove this by mathematical induction.
Base case:
a0 = 2 = 2(1+0)
a1 = 4 = 2(1+1)
Induction hypothesis: Assume that an = 2(1+n) for some n.
Induction step:
an+1 = an + 2^(n+1) (given recurrence relation)
= 2(1+n) + 2^(n+1) (using induction hypothesis)
= 2 + 2n + 2^n * 2
= 2 + 2(n+1)
= 2(1+(n+1))
Therefore, the induction step is true and the pattern an = 2(1+n) holds for all n.
3. Conclusion
Hence, the solution to the recurrence relation an= an-1 2n, with initial term a0= 2 is option 'B' i.e., 2(1+n).
Given recurrence relation is an= an-1 2n, with initial term a0= 2. We need to find the solution for this recurrence relation.
1. Finding the first few terms of the sequence
Let's find the first few terms of the sequence using the recurrence relation.
a0 = 2 (given)
a1 = a0 + 2^1 = 2 + 2 = 4
a2 = a1 + 2^2 = 4 + 4 = 8
a3 = a2 + 2^3 = 8 + 8 = 16
a4 = a3 + 2^4 = 16 + 16 = 32
2. Observing the pattern
Looking at the first few terms, we can observe that an = 2(1+n). Let's prove this by mathematical induction.
Base case:
a0 = 2 = 2(1+0)
a1 = 4 = 2(1+1)
Induction hypothesis: Assume that an = 2(1+n) for some n.
Induction step:
an+1 = an + 2^(n+1) (given recurrence relation)
= 2(1+n) + 2^(n+1) (using induction hypothesis)
= 2 + 2n + 2^n * 2
= 2 + 2(n+1)
= 2(1+(n+1))
Therefore, the induction step is true and the pattern an = 2(1+n) holds for all n.
3. Conclusion
Hence, the solution to the recurrence relation an= an-1 2n, with initial term a0= 2 is option 'B' i.e., 2(1+n).
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Community Answer
The solution to the recurrence relation an= an-1+ 2n, with initial ter...
When n = 1, a1 = a0 + 2. By substitution we get, a2 = a1 + 2 ⇒ a2 = (a0 + 2)+2 and so on. So the solution to the recurrence relation, subject to the initial condition should be an = 2 + 2n = 2(1+n).
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