then 9tm+1 isalso triangular.Write each of the following numbers as the sum ofthree or fewer triangular numbers:169,(b)(c) 185,(a) 56,(d) 287.5. For n1, establish the formula(2n+1)?1(4t,+1)-(4t,2BUS6Verify that 1225 and 41,616 are simultaneously squareand triangular numbers. [Hint: Finding an integer nsuch thatgtinidmoubonincb ofnon(n + 1)dutn12252udis equivalent to solving the quadratic equationn2n- 2450 = 0.]*w7. An oblong number counts the number of dots in arectangular array having one more row than it hascolumns; the first few of these numbers aresln01 202 6 03 1204 =20and in general, the nth oblong number is given byОnn(n+1). Prove algebraically and geometricallythat(а)On 2+4 6+ + 2n.Any oblong number is the sum of two equal11(): :

Question
Asked Sep 24, 2019
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Number 6 please. 

then 9tm+1 is
also triangular.
Write each of the following numbers as the sum of
three or fewer triangular numbers:
1
69,
(b)
(c) 185,
(a) 56,
(d) 287.
5. For n
1, establish the formula
(2n+1)?
1
(4t,+1)-(4t,2
BUS
6
Verify that 1225 and 41,616 are simultaneously square
and triangular numbers. [Hint: Finding an integer n
such that
g
tinidmou
bonincb of
no
n(n + 1)
dutn
1225
2
ud
is equivalent to solving the quadratic equation
n2n- 2450 = 0.]
*w
7. An oblong number counts the number of dots in a
rectangular array having one more row than it has
columns; the first few of these numbers are
sln
01 2
02 6 03 12
04 =20
and in general, the nth oblong number is given by
Оn
n(n+1). Prove algebraically and geometrically
that
(а)
On 2+4 6+ + 2n.
Any oblong number is the sum of two equal
11
()
: :
help_outline

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then 9tm+1 is also triangular. Write each of the following numbers as the sum of three or fewer triangular numbers: 1 69, (b) (c) 185, (a) 56, (d) 287. 5. For n 1, establish the formula (2n+1)? 1 (4t,+1)-(4t,2 BUS 6 Verify that 1225 and 41,616 are simultaneously square and triangular numbers. [Hint: Finding an integer n such that g tinidmou bonincb of no n(n + 1) dutn 1225 2 ud is equivalent to solving the quadratic equation n2n- 2450 = 0.] *w 7. An oblong number counts the number of dots in a rectangular array having one more row than it has columns; the first few of these numbers are sln 01 2 02 6 03 12 04 =20 and in general, the nth oblong number is given by Оn n(n+1). Prove algebraically and geometrically that (а) On 2+4 6+ + 2n. Any oblong number is the sum of two equal 11 () : :

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Expert Answer

Step 1

Problem 6:

Recall the following facts.

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Square number: N is a square number if there exists an integer s such that s2 = N Triangle number: T is a triangle number if there exists an natural number n such that "(n+1) =T 2 n2 2T 0 is solvable in N

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Step 2

Verify 1225 is simultaneously square an...

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1225 is a square number Since there exists an integer 35 such that 352 1225 1225 is a triangle number n22(1225 0 nn-2450 0 (n-49) (n50) =0 n-50,49 n 49 e N, natural number Thus n2 2(1225) 0 is solvable in natural numbers

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