Hi,
please could someone assist with the following:
This is the pattern give:
Pattern number 1 2 3 4
Number of dots 2 6 12 20
The question is: Find s formula for the general term
Their solution goes as follows:
The second difference is constant therefore Tn is quadratic - I understand this
therefore Tn = an2+bn+c this i understand
but 2a=a - I cannot understand the basis of this statement
Can anyone enlighten me?
Thanks!
You say you understand that . Since you have 3 numbers to determine, a, b, and c, you need 3 equations. Pick any 3 numbers from the sequence.
Since , .
since ,
Since ,
Subtract the first equation from the second to get (4a+ 2b+ c)- (a+ b+ c)= 3a+ b= 6- 2= 4 or 3a+ b= 4. c has been eliminated.
Subtract the first equation from the third to get (8a+ 3b+ c)- (a+ b+ c)= 7a+ 2b= 12- 2= 10 or 7a+ 2b= 10. Again c has been eliminated. Now we have the two equations 3a+ b= 4 and 7a+ 2b= 10.
Multiply 3a+ b= 4 by 2 to get 6a+ 2b= 8 and subtract that from 7a+ 2b= 10: (7a+ 2b)- (6a+ 2b)= a= 10- 8= 2.
a= 2, not "2a= a".
Now that you know a= 2, use either 3a+ b= 4 or 7a+ 2b= 10 to determine b. Then use any of the original 3 equations to determine c.
As a check, see if those values for a, b, and c give .