ll
Try this for n = 1:
1*3 =? 1(1 + 1)(2*1 +7)/6
3 =? 1*2*9/6
3 =? 3 (Check!)
Now assume the statement is true for some integer k. That is
1*3 + 2*4 + ... + k(k + 2) = k(k + 1)(2k + 7)/6
We need to show that it also holds for k + 1:
1*3 + 2*4 + ... + k(k + 2) + (k + 1)(k + 3) = (k + 1)(k + 2)(2k + 9)/6
We know that 1*3 + ... + k(k + 2) = k(k + 1)(2k + 7)/6 so use that in the above equation. Thus
k(k + 1)(2k + 7)/6 + (k + 1)(k + 3) = (k + 1)(k + 2)(2k + 9)/6
should be an identity.
Multiply both sides by 6, then simplify:
k(k + 1)(2k + 7) + 6(k + 1)(k + 3) = (k + 1)(k + 2)(2k + 9)
(k^2 + k)(2k + 7) + 6(k^2 + 4k + 3) = (k^2 + 3k + 2)(2k + 9)
2k^3 + 9k^2 + 7k + 6k^2 + 24k + 18 = 2k^3 + 15k^2 + 31k + 18
2k^2 + 15k^2 + 31k + 18 = 2k^3 + 15k^2 + 31k + 18
0 = 0 (Check!)
Thus the proposition is true for k = 1, so it is true for all integer k >= 1.
-Dan
Let P(n) be the proposition: 57 divides 7^(n+2) + 8^(2n+1).
(a) P(1) is true as 7^3+8^3 = 855 = 15*57
(b) Now suppose for some k>=1, P(k) is true. Then 7^(k+2) + 8^(2k+1) = a*57
for some positive integer a, or 8^(2k+1) = a*57 - 7^(k+2).
Now:
7^((k+1)+2) + 8^(2(k+1)+1) = 7*7^(k+2) + 64*8^(2k+1)
........................................= 7*7^(k+2) + 64*[a*57 - 7^(k+2)]
........................................= [7 - 64]* 7^(k+2) + 64*a*57
........................................= 57*[ -7^(k+2) + 64*a]
Hence 7^((k+1)+2) + 8^(2(k+1)+1) is divisible by 57, which proves P(k+1),
therfore with (a) this proves P(n) is true for all n>=1 by mathematical
induction.
RonL
Hello, chillerbros17!
Another approach to #1 . . .
1) Prove by Mathematical Induction:
. . 1·3 + 2·4 + 3·5 + . . . + n(n + 2) .= .n(n + 1)(2n + 7)/6 .for all integers > 1
Verify S(1): .1·3 .= .1(2)(9)/6 . → . 3 = 3 . . . true!
Assume S(k) is true: .1·3 + 2·4 + 3·5 + ... + k(k + 2) .= .k(k + 1)(2k + 7)/6
We want to prove that S(k+1) is true:
. . 1·3 + 2·4 + 3·5 + ... + (k + 1)(k + 3) .= .(k + 1)(k + 2)(2[k+1] + 7)/6 .[1]
written out here for reference ... so we'll recognize it later.
Begin with S(k): .1·2 + 2·4 + 3·5 + ... + k(k + 2) .= .k(k + 1)(2k + 7)/6
Add (k + 1)(k + 3) to both sides:
. . 1·3 + 2·4 + 3·5 + ... + (k + 1)(k + 3) .= .k(k + 1)(2k + 7)/6 + (k + 1)(k + 3)
The left side is the left side of [1].
We must show that the right side is the right side of [1].
. . . . . . . . . . . . . k(k + 1)(2k + 7) . . . . . . . . . . . . . . . k(k + 1)(2k + 7) + 6(k + 1)(k + 3)
The right side is: .-------------------- + (k + 1)(k + 3) .= . -----------------------------------------
. . . . . . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
. . . . . . .(k + 1) [k(2k + 7) + 6(k + 3)] . . . (k + 1)(2kČ + 13k + 18)
Factor: . ------------------------------------ .= .-----------------------------
. . . . . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . . . . 6
. . . . . (k + 1)(k + 2)(2k + 9) . . .(k + 1)(k + 2)(2[k+1] + 7)
. . . = .-------------------------- .= .--------------------------------
. . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . . . 6
This is the right side of [1] . . . The induction proof is complete.