Respuesta :
Answer:
The n-th term for the sequence will be: [tex]n^2+4n+2[/tex]
Step-by-step explanation:
Given sequence is: 7, 14, 23, 34, 47. 62, 79, ........
The n-th term of a quadratic sequence is: [tex]t_{n}=an^2 +bn+c[/tex]
For [tex]n=1[/tex]....
[tex]t_{1}=a(1)^2+b(1)+c\\ \\ a+b+c=7 .............................(1)[/tex]
For [tex]n=2[/tex]....
[tex]t_{2}=a(2)^2+b(2)+c\\ \\ 4a+2b+c=14 .............................(2)[/tex]
For [tex]n=3[/tex]....
[tex]t_{3}=a(3)^2+b(3)+c\\ \\ 9a+3b+c=23 .............................(3)[/tex]
Subtracting equation (1) from equation (2), we will get......
[tex]3a+b=7..........................(4)[/tex]
Subtracting equation (2) from equation (3), we will get.......
[tex]5a+b=9..........................(5)[/tex]
Now, subtracting equation (4) from equation (5)...........
[tex]2a=2\\ \\ a=\frac{2}{2}=1[/tex]
Plugging this [tex]a=1[/tex] into equation (4), we will get....
[tex]3(1)+b=7\\ \\ 3+b=7\\ \\ b=7-3=4[/tex]
Now, plugging [tex]a=1[/tex] and [tex]b=4[/tex] into equation (1) .........
[tex]1+4+c=7\\ \\ 5+c=7\\ \\ c=7-5=2[/tex]
Thus, the n-th term for the sequence will be: [tex]n^2+4n+2[/tex]
