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

Observe the following pattern and fill in the missing number. 
$ \displaystyle 11^{2} =121$
$ \displaystyle 101^{2} =10201$
$ \displaystyle 10101^{2} =102030201$
$ \displaystyle 1010101^{2} =......................$

  1. $ \displaystyle 1010101^{2} $=10203030201
  2. $ \displaystyle 1010101^{2} $=10204040201
  3. $ \displaystyle 1010101^{2} $=1020304030201
  4. $ \displaystyle 1010101^{2} $=10204030201
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$11^{2}$$=$$121$

$101^{2}$$=$$10201$
$10101^{2}$$=$$10203020101$
$1010101^{2}$$=$$1020304030201$
$101010101^{2}$$=$$10203040504030201$
We will go up to number of  ones in the number numerically.
Hence, Option C is correct.

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

Find the sum of:
$1+3+5+7+9+11+13+15+17+19+21+23$

  1. $11^2$
  2. $12^2$
  3. $10^2$
  4. $13^2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$S _{n}=\dfrac{n}{2}\left [ 2a+(n-1)d \right ]$
$a =$ First term
$d =$ Common difference
$n =$ number of terms

Given series is an A.P.
with first term $=$ $1$
common difference $=$ $2$
and last term $=$ $23$

Sum $=$ $\dfrac{12}{2}$ $\times$ $(2+11\cdot 2)$ $=$ $12\times12$
Hence, Option B is correct.