Let ⟨an⟩ be a sequence such that a0=0, a1=21 and 2an+2=5an+1−3an, n=0,1,2,3,…. Then ∑k=1100ak is equal to
Get the complete, step-by-step math solution for: "Let a_n be a sequence such that a_0 = 0, a_1 = (1)/(2) and 2 a_{n+2} = 5 a_{n+1} - 3 a_n, n = 0, 1, 2, 3, . Then _{k=1}^{100} a_k is equal to". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Find the characteristic equation
The given recurrence relation is a linear homogeneous recurrence relation with constant coefficients. To find the general form of an, we first write its characteristic equation by replacing an+2 with r2, an+1 with r, and an with 1.
Step 2: Solve the characteristic equation for roots
We solve the quadratic characteristic equation to find its roots. Factoring the quadratic equation 2r2−5r+3=0 gives us (2r−3)(r−1)=0. This yields two distinct roots, r1=23 and r2=1.
Step 3: Determine the general form of an
Since the roots are distinct, the general form of the sequence an is Ar1n+Br2n, where A and B are constants. Substituting the roots, we get an=A(23)n+B(1)n.
Step 4: Use initial conditions to find A and B
We use the given initial conditions a0=0 and a1=21 to find the values of A and B.
For n=0: a0=A(3/2)0+B(1)0=A+B=0⟹B=−A.
For n=1: a1=A(3/2)1+B(1)1=23A+B=21.
Substituting B=−A into the second equation: 23A−A=21⟹21A=21⟹A=1.
Then B=−1. Thus, an=(23)n−1.
Step 5: Calculate the sum of the sequence
We need to find the sum of the first 100 terms of the sequence, starting from k=1. We can split the sum into two parts: a geometric series and a sum of constants. ∑k=1100(23)k−∑k=11001 The first part is a geometric series with first term a=23, common ratio r=23, and n=100 terms. The sum of a geometric series is Sn=ar−1rn−1.
So, S100=2323−1(23)100−1=2321(23)100−1=3[(23)100−1].
The second part is ∑k=11001=100.
Therefore, the total sum is 3[(23)100−1]−100.
Step 6: Simplify the sum
Simplifying the expression obtained in the previous step, we distribute the 3 and combine the constant terms.