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

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

Haider Ali Dar

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