Question

A statement Sn about the positive integers is given. Write statements S1 S2 and S3 simplifying Sk+1 completely. Sn : 1+ 4 +7 +......+(3n-2)= n(3n-1)/2

Answer 1

Sn:1+4+7+....(3n-2)=n(3n-1)/2
S1:1=1(3*1-1)/2=1
S2: 1+4=2(3*2-1)/2
or5=5
S3: 1+4+7=3(3*3-1)/2
or 12=3*8/2=12
Assume Sk is true for all positive integers.
1+4+7+.....+(3k-2)=k(3k-1)/2
adding both sides 3(k+1)-2
1+4+7+...+(3k-2)+{3(k+1)-2}=k(3k-1)/2 +3(k+1)-2
=(3k^2-k+6k+6-4)/2
=(3k^2+5k+2)/2
=(3k^2+3k+2k+2)/2
={3k(k+1)+2(k+1)}/2
=(k+1)(3k+2)/2
=(k+1){3(k+1)-1}/2
or Sk+1=(k+1){3(k+1)-1}/2

Answer 2

Thank you

Answer 3

yw