Question

При яких натуральних значеннях n число 2n+65 є квадратом цілого числа?

Answer 1

9²=81      ,  2n+65=81   ⇒   2n=16  ,  n=8
10²=100  ,  2n+65=100   ⇒   2n=35  ,  35 не делится на 2
11²=121  ,  2n+65=121   ⇒   2n=56  ,  n=28
12²=144  ,  2n+65=144   ⇒   2n=79  ,  79 не делится на 2
13²=169  ,  2n+64=169   ⇒   2n=104 ,  n=52
................................................................
n=8 , 28 , 52 , ......

Answer 2

Natural Values of n for 2n + 65 to Be a Perfect Square

To find the natural values of n for which the number 2n + 65 is a perfect square, we can use the following approach:

Step 1: Express 2n + 65 as a perfect square: 2n + 65 = m^2, where m is an integer.

Step 2: Solve for n in terms of m: n = (m^2 - 65) / 2.

Step 3: Determine the natural values of n by finding the natural values of m for which the expression (m^2 - 65) / 2 results in a natural number.

Let's proceed with the calculations to find the natural values of n for which 2n + 65 is a perfect square.