Question

A1=1 An+1= An+3 И A1=2 An+1=3An Найти А5 рекуррентно

Answer 1

$1)\; \; a_1=1\; ,\; \; a_{n+1}=a_{n}+3\\\\a_2=a_1+3=1+3=4\; ,\\\\a_3=a_2+3=4+3=7\\\\a_4=a_3+3=7+3=10\\\\a_5=a_4+3=10+3=13\\\\ili\\\\a_5=a_4+3=(a_3+3)+3=a_3+2\cdot 3=(a_2+3)+2\cdot 3=\\\\=a_2+3\cdot 3=(a_1+3)+3\cdot 3=a_1+4\cdot 3=1+4\cdot 3=13\\\\\\2)\; \; a_1=2\; ,\; \; a_{n+1}=3\, a_{n}\\\\a_2=3\, a_1=3\cdot 2=6\\\\a_3=3\, a_2=3\cdot 6=18\\\\a_4=3\, a_3=3\cdot 18=54\\\\a_5=3\, a_4=3\cdot 54=162$

$ili\\\\a_5=3\, a_4=3\cdot (3\, a_3)=3^2\cdot a_3=3^2\cdot (3\, a_2)=3^3\cdot a_2=\\\\=3^3\cdot (3\, a_1)=3^4\cdot a_1=3^4\cdot 2=81\cdot 2=162$

Answer 2

To find the value of A5 recursively using the given sequences, we can use the formulas provided:

Sequence 1: A1 = 1 and An+1 = An + 3 Sequence 2: A1 = 2 and An+1 = 3An

Let's calculate the values step by step:

Using Sequence 1: A1 = 1 A2 = A1 + 3 = 1 + 3 = 4 A3 = A2 + 3 = 4 + 3 = 7 A4 = A3 + 3 = 7 + 3 = 10 A5 = A4 + 3 = 10 + 3 = 13

Using Sequence 2: A1 = 2 A2 = 3A1 = 3 * 2 = 6 A3 = 3A2 = 3 * 6 = 18 A4 = 3A3 = 3 * 18 = 54 A5 = 3A4 = 3 * 54 = 162

Therefore, A5 can be calculated recursively using either sequence as:

A5 = 13 (using Sequence 1) or A5 = 162 (using Sequence 2)