Here are the answers worked out for the sample test!
 
term think pattern
1 2 2
2 2+3 2+3(1)
3 5+6 5+3(2)
4 11+9 11+3(3)
5 20+12 20+3(4)
6 32+15 32+3(5)
n
tn-1+3(n-1)
 
1 + 4 + 9 + 16 + . . . + k2 = k(k + 1)(2k + 1)/6 is true.
1 + 4 + 9 + . . . + k2 + (k + 1)2 = (k+1)(k+1+1)(2k+2+1)/6
replace the sequence in red with what you assumed to be true in step 2.
k(k+1)(2k+1)/6 +(k+1)2 = (k+1)(k+2)(2k+3)/6
Working on the left side yields:
k(k+1)(2k+1)/6 + (6k2 + 12k + 6)/6 =
[(k2 + k)(2k+1) + (6k2 + 12k + 6)]/6 =
[(2k3 + 3k2 + k) + (6k2 + 12k + 6)]/6 =
(2k3 + 9k2 + 13k + 6)/6 =
Working on the right side gives us:
= [(k2 + 3k + 2)(2k + 3)]/6
distributing yields:
= (2k3 + 9k2 + 13k + 6)/6
Therfore, it works!!!!


Congratulations!! You have finished chapter 13!
Next chapter is chapter 19!!
Can't wait!!
Bye!