Tag: triangular numbers

Questions Related to triangular numbers

Evaluate $14 \times 16$ using even-even pattern.

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

  1. 354

  2. 244

  3. 444

  4. 224

Show answer
Correct Option: D
Explanation:

$14\times 16=(10+4)$$\times$$(10+6)$ 

$=$ $100+60+40+24$ $=$  $224$
Hence, Option D is correct.

Having $5$ at units place, find the square of the number $185$.

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

  1. $34225$

  2. $48034$

  3. $15620$

  4. $83450$

Show answer
Correct Option: A
Explanation:

$185$ $=$ $(200-15)$

$185^{2}$ $=$ $(200-15)^{2}$
$(200-15)^{2}$ $=$ $200^{2}$ $+$ $15^{2}$ $-$ $2\times200\times15$
$185^{2}$ $=$ $40000$ $+$ $225$ $-$$6000$
$185^{2}$ $=$ $34225$
Hence, Option A is correct.

Evaluate $31 \times 33$ using odd-odd pattern.

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

  1. $423$

  2. $823$

  3. $923$

  4. $1023$

Show answer
Correct Option: D
Explanation:

$31\times 33=(30+1)$$\times$$(30+3)$ 

$=$ $900+90+30+3$ $=$  $1023$
Hence, Option D is correct.

Evaluate: $11^2$

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

  1. $131$

  2. $60+61$

  3. $141$

  4. None of these

Show answer
Correct Option: B
Explanation:

$11^{2}$ $=$ $(10+1)^{2}$ $=$ $100+1+20$

$=121$ $=$ $60+61$
Hence, Option B is correct.

Find the value of $15^2$.

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

  1. 224

  2. 125

  3. 112+113

  4. None of these

Show answer
Correct Option: C
Explanation:

$15^{2}$ $=$ $(10+5)^{2}$ $=$ $100+25+100$

$=225$ $=$ $112+113$
Hence, Option C is correct.

Without adding the numbers, find the sum of 1 + 3 + 5 + 7.

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

  1. 16

  2. 15

  3. 14

  4. 12

Show answer
Correct Option: A
Explanation:

There are 4 consecutive odd numbers.
We know the formula for consecutive odd numbers of their sum = $n^2$
So, Sum = $4^2$ = 16
So, 1 + 3 + 5 + 7 = 16.

The sum of first 13 consecutive odd numbers is ___.

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

  1. 196

  2. 169

  3. 13

  4. 81

Show answer
Correct Option: B
Explanation:

We know the formula for consecutive odd numbers of their sum $= n^2$
So, Sum of $13$ consecutive odd numbers $= 13^2 = 169$

What is the series and also find the total of first $100$ consecutive odd numbers?

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

  1. $1 + 2 + 4 + 6 + 8 + 10 +... 100 = 12000$

  2. $2 + 3 + 4 + 7 + 9 + 11 +...100 = 10000$

  3. $1 + 3 + 5 + 7 + 10 + 11 +....100 = 1000$

  4. $1 + 3 + 5 + 7 + 9 + 11 +...100 = 10000$

Show answer
Correct Option: D
Explanation:

Sum of first $100$ consecutive odd numbers $= 1 + 3 + 5 + 7 + 9 + 11 + ....+100 = 10000$
Since, we know the formula for sum of consecutive odd numbers $= n^2$
So, $n = 100$, Sum $= 100^2 = 10000$

Find the sum of two consecutive number for $13^2$.

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

  1. $84$ and $85$

  2. $83$ and $84$

  3. $86$ and $82$

  4. $81$ and $80$

Show answer
Correct Option: A
Explanation:

Let the two consecutive numbers be, $\dfrac{n^{2} - 1}{2}$ and $\dfrac{n^{2} + 1}{2}$
$13^2= 169$
n = 13
$\dfrac{n^{2} - 1}{2} = \dfrac{13^{2} - 1}{2} = 84$
$\dfrac{n^{2} + 1}{2} = \dfrac{13^{2} + 1}{2} = 85$
So, the sum of two consecutive numbers = 84 + 85 = 169.

$21^{2}-1$ is a product of two consecutive even numbers. Find those numbers.

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

  1. 21 and 22

  2. 22 and 24

  3. 20 and 22

  4. 22 and 23

Show answer
Correct Option: C
Explanation:

$21^{2}-1 = 400$
we can express the above number into general form $a^{2}-1 = (a + 1)\times (a - 1)$
Where a = 21
So, $21^{2}-1 = (21 + 1)\times (21 - 1)$
= $22 \times 20 = 400$
Therefore, the two even consecutive numbers are 20 and 22.