Mathematics: For the sequence given recursively by ak = 2ak-1 - 3 for k 2, where a1 = 4, someone has guessed a closed form and used the predicate r(n) for the statement an = 2^n-1 +3. an = 23. oro s (a) are r(1), r(2), r(3), and r(4) true?

The answer is 500Step-by-step explanation:...

For the sequence given recursively by ak = 2ak-1

sweetLips230

04.01.2022

True

Step-by-step explanation:

Given that recurrence formula is

$a_k = 2a_{k-1} -3\\,k2\\$ and

a_1 =4

From this we can find

$a_2 = 2a_1-3\\= 8-3 =5\\a_3 =2(5)-3 =7\\a_4 = 2(7)-3 = 11\\$

Let us check now R's for these numbers

R(1) is trivially true

$R(2) = 2^{2-1} +3 =5\\R(3) = 2^{3-1} +3 =7\\R(4) = 2^{4-1} +3 =11$

We find that these matches with a values we found using recurrence formula

Hence we can say that R(1), R(2), R(3), and R(4) are true

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