Answer: The number that belongs to both sequences is 26.
Step-by-step explanation:
We have two sequences, let's call one as A and the other as B.
The n-th term of sequence A is written as:
aₙ = 3*n^2 - 1
the nth term of sequence B is written as:
bₙ = 30 - n^2
We want to find a term that belongs to both sequences, (it can be for different integers, we can use n for sequence A and x for sequence B)
Then we want to find:
aₙ = bₓ
where n and x are integer numbers.
Then we will heave:
3*n^2 - 1 = 30 - x^2
To find the pair, we could isolate one of the variables, then input different integers in the other variable and see if the outcome is also an integer.
Let's isolate n.
3*n^2 = 30 - x^2 + 1
3*n^2 = 31 - x^2
n^2 = (31 - x^2)/3
n = √( (31 - x^2)/3)
Now let's input different values for x, and see if the outcome is also an integer, notice that x is in a negative term inside a square root, then we have only a few values of x such that the equation can be true.
Then let's start with x = 1.
n(1) = √( (31 - 1^2)/3) = √(30/3) = √10
We know that √10 is not an integer.
now with x = 2,
n(2) = √( (31 - 2^2)/3) = √( (31 - 4)/3) = √(27/3) = √9 = 3
then if x = 2, we have n = 3.
Both of them are integers, then we get:
a₂ = 3*(3)^2 - 1 = 27 - 1 = 26
b₃ = 30 - 2^2 = 30 - 4 = 26
The number that belongs to both sequences is 26.
°