Asked by allexelle
A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely.
Sn: 1 + 4 + 7 + . . . + (3n - 2) = n(3n - 1)/2
please, can you help me I need the formula how to solve it.
Sn: 1 + 4 + 7 + . . . + (3n - 2) = n(3n - 1)/2
please, can you help me I need the formula how to solve it.
Answers
Answered by
Steve
well, just plug in k and k+1 for n.
You need to prove that if S<sub>k</sub> is true, then S<sub>k+1</sub> is also true. Then, if you can show that S1 is true, then S2,S3,... are all true as well.
1+4+...+(3k-2)+(3(k+1)-2) = S<sub>k</sub>+3(k+1)-2
= k(3k-1)/2 + 3(k+1)-2
= (k(3k-1) + 6(k+1)-4)/2
= (3k^2-k+6k+6-4)/2
= (3k^2+5k+2)/2
= (k+1)(3k+2)/2
= (k+1)(3(k+1)-1)/2
= S<sub>k+1</sub>
You need to prove that if S<sub>k</sub> is true, then S<sub>k+1</sub> is also true. Then, if you can show that S1 is true, then S2,S3,... are all true as well.
1+4+...+(3k-2)+(3(k+1)-2) = S<sub>k</sub>+3(k+1)-2
= k(3k-1)/2 + 3(k+1)-2
= (k(3k-1) + 6(k+1)-4)/2
= (3k^2-k+6k+6-4)/2
= (3k^2+5k+2)/2
= (k+1)(3k+2)/2
= (k+1)(3(k+1)-1)/2
= S<sub>k+1</sub>
Answered by
allexelle
thank you