Question

Recursion let a(n) = a(n-1) + 2*a(n-2) + 2 a(0) = 1, a(1) = 0. the value of a(15) will be


a. 32766


b. 16383


c. 65532


d. 32761 what is the answer

Answer 1

32761what is the answer

Answer 2

Answer:

$a_{15}=16384(2)-2=32766$

A is correct.

Step-by-step explanation:

We are given a recursion formula  

$a_n=a_{n-1}+2a_{n-2}+2$

$a_{0}=1,a_1=0$

$Put\ n=2, a_2=a_1+2a_0+2\Rightarrow 0+2(1)+2\Rightarrow 4$

$Put\ n=3, a_3=a_2+2a_1+2\Rightarrow 4+2(0)+2\Rightarrow 6$

$Put\ n=4, a_4=a_3+2a_2+2\Rightarrow 6+2(4)+2\Rightarrow 16$

$Put\ n=5, a_5=a_4+2a_3+2\Rightarrow 16+2(6)+2\Rightarrow 30$

$Put\ n=6, a_6=a_5+2a_4+2\Rightarrow 30+2(16)+2\Rightarrow 64$

$Put\ n=7, a_7=a_6+2a_5+2\Rightarrow 64+2(30)+2\Rightarrow 126$

$Put\ n=8, a_8=a_7+2a_6+2\Rightarrow 126+2(64)+2\Rightarrow 256$

$Put\ n=9, a_9=256(2)-2=510$

$Put\ n=10, a_{10}=510(2)+4=1024$

$Put\ n=11, a_{11}=1024(2)-2=2046$

$Put\ n=12, a_{12}=2046(2)+4=4096$

$Put\ n=13, a_{13}=4096(2)-2=8190$

$Put\ n=14, a_{14}=8190(2)+4=16384$

$Put\ n=15, a_{15}=16384(2)-2=32766$

Hence, A is correct.