**Prove by induction that for all positive integers
n.**

**Answer must follow the steps below:**

1- Clearly state P(n)

2- Basis of induction: Chose an appropriate n0 and prove that P(n0) is true.

3- Induction Step: Prove that P(k) --> P(k+1) is true for all k>= N0

4- Conclusion

Community Answer

1)P(n):for positive integer n, sum_(i=1)^(n)(-1)^(i)i^(2)=((-1)^(n)n(n+1))/(2)   2) claim p(1) is true, put n=1 sum_(i=1)^(1)(-1)^(i)i^(2)=(-1)^(1)1^(2)=-1 u200bu200bu200bu200b((-1)^(n)n(n+1))/(2)=((-1)^(1)1(1+1))/(2)=-1   sum_(i=1)^(1)(-1)^(i)i^(2)=((-1)^(1)1(1+1))/(2) Now p(1) is true.    3)assume that p(k) is true. We have to prove that that p(k+1) is true.  P(k) is true implies.  sum_(i=1)^(k)(-1)^(i)i^(2)=((-1)^(k)k(k+1))/(2) And {:[sum_(i=1 ... See the full answer