Tag: triangular numbers

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Multiple choice maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

Fill in the blanks:
$10^2 +1^2 + 10^2 = 10^2$
$12^2 + 2^2 + 6^2 = 12^2$
$14^2 + 7^2$ + ____ = ____

  1. $3^2, 14^2$
  2. $2^2, 14^2$
  3. $2^2, 7^2$
  4. $2^2, 12^2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

From the pattern, the third number is the division of the first two number.
The fourth number can be obtained by same as the first number.
Then, the missing numbers will be
$14^2 + 7^2 + 2^2 = 14^2$
So, $2^2, 14^2$ are the missing numbers.

Multiple choice maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

Which statement is true about consecutive natural numbers?

  1. The numbers between the difference of square of consecutive numbers is $2n + 1$
  2. The non-perfect square numbers between the square of consecutive numbers is $2n$

  3. The sum of the squares of two consecutive numbers is never a perfect square
  4. $n^{2} - 1$ is the standard form of the difference between two consecutive numbers
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$1^{2}$$=$$1$

$2^{2}$$=$$4$
$3^{2}$$=$$9$
$4^{2}$$=$$16$
$5^{2}$$=$$25$
$6^{2}$$=$$36$
$7^{2}$$=$$49$
Between $1$ and $4$, there are 2 numbers that is $2$ and $3$.
Between $4$ and $9$, there are 4 numbers that is $5$,$6$,$7$ and $8$.
So, there are $2n$ numbers between square of consecutive numbers where $n$is the smaller number.
Hence, Option B is correct.


Multiple choice maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

Let $S$ be the set of all ordered pairs $(x,y) $ of positive integers satisfying the condition $x^{2}-y^{2}=12345678$. Then:

  1. $S$ is an infinite set
  2. $S$ is the empty set
  3. $S$ has exactly one element
  4. $S$ is a finite set and has at least two elements
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$x^{2}-y^{2}=12345678 (x,y \, \epsilon \,  1^{+})$

RHS is even, so, x & y should be odd integer but difference square of two odd integer is multiple of 8 but RHS is not multiple of $8$

$\therefore  8 $ is an empty set.
Multiple choice maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

If a number of $n$-digits is perfect square and $n$ is an odd number, then which of the following is the number of digits of its square root?

  1. $\cfrac{n-1}{2}$
  2. $\cfrac{n}{2}$
  3. $\cfrac{n+1}{2}$
  4. $2n$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

no of digits in a perfect square is $n$

 If $n$ is odd then no of digits in its square roots is $\dfrac{n+1}{2}$