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If the \(n\)th term of a quadratic sequence is known, we can work out the terms in that sequence and also use knowledge of solving equations to determine if a specified number is a term in the sequence.

Write the first 5 terms of the sequence \(2n^2 + 1\)

\(n\) represents the position in the sequence. The first term in the sequence is when \(n = 1\), the second term in the sequence is when \(n = 2\), and so on.

To find the terms, substitute the position number for \(n\).

1st term: \(n = 1\)

2nd term: \(n = 2\)

3rd term: \(n = 3\)

4th term: \(n = 4\)

5th term: \(n = 5\)

So the first 5 terms of the sequence \(2n^2 + 1\) are 3, 9, 19, 33, 51.

When the \(n^{th}\) term is known, it can be used to work out specific terms in a sequence.

What is the 50th term in the sequence \(2n^2 + 7\)?

\(n = 50\) for the 50th term, so to find this, we need to substitute 50 in place of \(n\).

\(2n^2 + 7\) = (2 × 50²) + 7 = 5,007

Is the number 100 in the sequence \(4n^2 - 10\)?

To work out whether 100 is in the sequence, put the \(n^{th}\) term equal to the number and solve the equation.