# How do you find the nth term of the sequence: 3,12,25,42?

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### Haider Ali Dar

Extremely passionate in being able to teach students and see them progress.

To find the nth term of this sequence, the first step is to find the common difference between each term; the difference between 3 and 12 is 9 and then the difference between 25 and 12 is 13. The second step is to find the second difference which is between 13 and 9 which is 4. We have found the second difference of the sequence which is 4. We have noticed this is a quadratic sequence and the second difference is exactly 4. The nth term of a quadratic sequence is in the form an^2+bn+c. To find the value of a, we just half the second difference 4 and that means a=2. To find the value of b and c, we have to create simultaneous equations using the our initial sequence. Right now we have 2n^2+bn+c, sub n=1 into the equation 2(1)^2+b(1)+c =3. The nth term equals 3 because the first value of he sequence is 3. Our first equation is 2+b+c=3, b+c=1. Sub n=2 for our second equation 2(2)^2+b(2)+c=12 8+2b+c=12 2b+c=4 (our second equation) Now solve the two equations simultaneously to work out our values of b and c. Solving them simultaneously we find the value of b= 3 and c-2 Therefore, the nth term of the sequence 3,12,25,42....is 2n^2+3n-2

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