Use mathematical induction to prove the following.1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + n(n + 1) = 
What will be an ideal response?
Answers may vary. One possibility:
Sn: 1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + n(n + 1) = 
S1: 1 ? 2 = 
Sk: 1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + k(k + 1) = 
Sk+1: 1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + k(k + 1) + (k + 1)(k + 2) = 
Step 1: Since 
= 
= 1 ? 2, S1 is true.
Step 2: Let k be any natural number. Assume Sk. Deduce Sk+1.
1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + k(k + 1) = 
1 ? 2 + 2 ? 3 + 3 ? 4 + . . . + k(k + 1) + (k + 1)(k + 2) = 
+ (k + 1)(k + 2)
= 
+ 
= 
= 
.
You might also like to view...
The magnitude of earthquake 1 is 4.5, and the magnitude of earthquake 2 is 6.5. How do their intensities compare? ?
A. Earthquake 1 is 100 times as intense as earthquake 2. B. Earthquake 1 is 2 times as intense as earthquake 2. C. Earthquake 2 is 2 times as intense as earthquake 1. D. Earthquake 2 is 100 times as intense as earthquake 1.
Solve the problem.The temperature at the South pole was -17° at 8 am. At 3 pm, it was 40°. By how many degrees did the temperature rise?
A. by 23° B. by -23° C. by -57° D. by 57°
Solve the problem.If a graph has 7 vertices and M is the number of vertices of odd degree, then M cannot be
A. 4. B. 7. C. 3. D. 0. E. none of these
Solve the problem.A private airplane leaves an airport and flies due east at 185 mph. Two hours later, a jet leaves the same airport and flies due east at 333 mph. When will the jet overtake the plane?
A. 1
hr
B. 
hr
C. 4
hr
D. 2
hr