Hello, harrietchemist!

I can help with the last two . . .

Let the first triangular number be: .T1 .= .k(k+1)/2Prove the following concerning triangular numbers:

c) the sum of any two consecutive triangular numbers is a perfect square

then the next triangular number is: .T2 .= .(k+1)(k+2)/2

Then: .T1 + T2 .= .k(k+1)/2 + (k+1)(k+2)/2 .= .(k² + k + k² + 3k + 2)/2

. . = .(2k² + 4k + 2)/2 .= .k² + 2k + 1 .= .(k + 1)² . . . a square

Since n = k(k+1)/2, then: .9n + 1 .= .9·k(k+1)/2 + 1d) if n is a triangular, then so are: 9n + 1, 25n + 3, and 49n + 6

. . = .(9k² + 9k + 2)/2 .= .(3k + 1)(3k + 2)/2 . . . a triangular number

Do the same for the other two expressions.