Let b0 b1 b2 be the sequence defined by the explicit formula...

Let b0, b1, b2 be the sequence defined by the explicit formula bn = C · 6n + D(−5)n for each integer n ≥ 0, where C and D are real numbers. Find C and D so that b0 = 0 and b1 = −11. What is b2 in this case? Find C and D so that b0 = 3 and b1 = 7. C=? D=? What is b2 in this case?

Transcript

00:01 In this problem we are given that the sequence b0 b1 b2 etc is defined explicitly by the formula bn is equal to c times 6 raise to n plus d times negative 5 the whole raise to n for each integer n greater than or equal to 0 where c and d are real numbers.

00:26 The first question is to determine the value of c and d so that b not is equal to zero and b1 is equal to negative 11 so we have the general equation is bn is equal to c times six raised to n plus d time negative 5 the whole raise to n so that b not is equal to zero implies when n is equal to zero we have zero is equal to c times six raised to zero plus d times negative 5 the whole raise to 0 that is from here we get the equation c plus t is equal to 0 mark it as equation 1 similarly b 1 is equal to negative 11 implies when n is equal to 1 we have negative 11 is equal to c times 6 plus d times negative 5 from which we get the equation 6c minus 5 d is equal to to negative 11 mark this as equation 2.

01:31 Now we can solve equations 1 and 2 simultaneously to get the value of c and d.

01:37 From this first equation we can multiply throughout by 6 to get 6c plus 6d is equal to 0.

01:47 Now from the second equation we can subtract this equation to get negative 11 d is equal to negative 11.

01:58 From which we get the value of d to be 1.

02:02 Now using this value of d and this equation 1, we have the value of c is negative 1.

02:10 Therefore, we get the value of c is negative 1 and the value of d is equal to 1 in this case.

02:16 These are the required first answers...

Question

Let b0, b1, b2 be the sequence defined by the explicit formula bn = C · 6n + D(−5)n for each integer n ≥ 0, where C and D are real numbers. Find C and D so that b0 = 0 and b1 = −11. What is b2 in this case? Find C and D so that b0 = 3 and b1 = 7. C=? D=? What is b2 in this case?

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