Tag: triangular numbers

Questions Related to triangular numbers

$11^{2}-1$ is a product of two consecutive even numbers. Find those two even numbers.

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

  1. 12 and 22

  2. 12 and 13

  3. 10 and 12

  4. 12 and 14

Show answer
Correct Option: C
Explanation:

$11^{2}-1 = 120$
we can express the above number into general form $a^{2}-1 = (a + 1)\times (a - 1)$
Where a = 11
So, $11^{2}-1 = (11 + 1)\times (11 - 1)$
= $12 \times 10 = 120$
Therefore, the two even consecutive numbers are 10 and 12.

Find the sum of two consecutive numbers for $15^2$.

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

  1. 112 and 113

  2. 113 and 114

  3. 115 and 112

  4. 113 and 115

Show answer
Correct Option: A
Explanation:

Let the two consecutive numbers be, $\dfrac{n^{2} - 1}{2}$ and $\dfrac{n^{2} + 1}{2}$
$15^2= 225$
n = 15
$\dfrac{n^{2} - 1}{2} = \dfrac{15^{2} - 1}{2} = 112$
$\dfrac{n^{2} + 1}{2} = \dfrac{15^{2} + 1}{2} = 113$
So, the sum of two consecutive numbers = 112 + 113 = 225.

$125^2$ is equal to the sum of one consecutive number 7813. Find the other.

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

  1. $7814$

  2. $7815$

  3. $7812$

  4. $7816$

Show answer
Correct Option: C
Explanation:

$125^2 = 7813 + ?$
$15625 - 7813 = 7812$
So, the other consecutive number is $7812.$

$96^{2}-1$ is a product of two consecutive odd numbers. Find those two odd numbers.

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

  1. 96 and 98

  2. 93 and 95

  3. 95 and 97

  4. 99 and 101

Show answer
Correct Option: C
Explanation:

$96^{2}-1 = 9215$
we can express the above number into general form $a^{2}-1 = (a + 1)\times (a - 1)$
Where a = 96
So, $96^{2}-1 = (96 + 1)\times (96 - 1)$
= $97 \times 95 = 9215$
Therefore, the two odd consecutive numbers are 95 and 97.

Find the series and also find the total of first 10 consecutive odd numbers.

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

  1. $1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100$

  2. $1 + 3 + 5 + 7 + 9 + 11 + 13 + 16 + 17 + 19 = 101$

  3. $1 + 3 + 5 + 7 + 10 + 11 + 13 + 15 + 17 + 19 = 101$

  4. $1 + 3 + 6 + 6 + 9 + 11 + 13 + 15 + 17 + 19 = 100$

Show answer
Correct Option: A
Explanation:

Sum of first 10 consecutive odd numbers $= 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100$
Since, we know the formula for sum of consecutive odd numbers = $n^2$
So, $n = 10$, Sum $= 10^2 = 100$

$25^2$ is equal to the sum of one consecutive number 313. Find the other.

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

  1. $311$

  2. $312$

  3. $314$

  4. $315$

Show answer
Correct Option: B
Explanation:

$25^2 = 313 + ?$
$625 - 313 = 312$
So, the other consecutive number is $312$.

Observe the following pattern and find the missing number.
$12^2 = 144$
$102^2 = 10404$
$1002^2 = 1004004$
$10000002^2 = ? $

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

  1. $100400400004$

  2. $100000040000004$

  3. $100040040004$

  4. $100404000004004$

Show answer
Correct Option: B
Explanation:

$12^2 = 144$
$102^2 = 10404$
$1002^2 = 1004004$
$10000002^2 = 100000040000004$
Start with 1 followed as many zeroes as there are between the first and the last 4, followed by two again followed by as many zeroes and end with 4.

Fill in the blanks:
$11^2 +8^2 + 3^2 = 19^2$
$12^2 + 2^2 + 10^2 = 14^2$
$14^2 + 7^2 $ + ____ = ____

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

  1. $7^2, 11^2$

  2. $14^2, 21^2$

  3. $7^2, 19^2$

  4. $7^2, 21^2$

Show answer
Correct Option: D
Explanation:

From the pattern, the third number is the difference of the first two numbers.
The fourth number can be obtained by addition of the first two numbers.
Then, the missing numbers will be
$14^2 + 7^2 + 7^2 = 21^2$
So, $7^2, 21^2$ are the missing numbers.

Find the missing number of the pattern.
$3^2 + 6^2 + 18^2 = 19^2$
$4^2 + 3^2 + 12^2$ = ___

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

  1. $13^2$

  2. $19^2$

  3. $7^2$

  4. $18^2$

Show answer
Correct Option: A
Explanation:

From the pattern, the third number is the product of the first two number.
The fourth number can be obtained by adding 1 to the third number.
Then, the missing number will be
$4^2 + 3^2 + 12^2 = 13^2$
So, $13^2$ is the missing number.

Find the missing number of the pattern.
$4^2 + 2^2 + 6^2 = 36^2$
$5^2 + 2^2 +$ ___ = $49^2$

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

  1. $13^2$

  2. $19^2$

  3. $7^2$

  4. $18^2$

Show answer
Correct Option: C
Explanation:

From the pattern, the third number is the sum of the first two number.
The fourth number can be obtained by squaring the third number.
Then, the missing number will be
$5^2 + 2^2 +7^2 = 49^2$
So, $7^2$ is the missing number.