Number Patterns

$786, 514, 378, 310, 276, ?$

If $a=b^2-b$, where $b$ is an integer then $a^2-2a$ is divisible by 

The largest common term to the sequences $1, 11, 21, 31, ....100$ terms and $31, 36, 41, 45, 46....100$ terms is

Identify the missing number $49,343,64,\dots ,81,729$

If $1^2 + 2^2 + 3^2 + ...... + {10}^2 = 385$, then the value of $2^2 + 4^2 + 6^2 + ..... + {20}^2$ will be _____.
 

The $5^{th}$ terms of the sequence defined by $t_1=2$, $t_2=3$, and $t_n=t_{n-1}+t_{n-2}$ for $n\ge 3$.

$2, 3, 13, 4, 5, 41, 6, 7, ?, ?$

$7+10+13+16+\dots$ is a sequence whereas $7,$ $10,$ $13,$ $16$ is a series.

Iman arranged $8$ number in two columns writing the numbers $1, 2, 3, 4, 5, 7, 8$ and $9$.

The other table to get the same sum from the columns by interchanging the numbers only in $2$ rows is

The correct option for finding $30$ using other one-digit number thrice except $\mathrm{3\; is}$

For a given A.P., $a=3.5, d=0$, then $t_n=$

Select the option from the given choices that completes the sequence by filling in the blank spaces.

$4, 9, 16, 25,$_____

Select the option from the given choices that completes the sequence by filling in the blank spaces.

$\mathrm{ABE}, \mathrm{BCF}, \mathrm{CDG}, ? , \mathrm{EFI}$

Identify the pattern and fill up the number in the blank.

$1, 8, 27, 64,$ _____

Identify the pattern and fill up the number in the blank.

$4, 2, 1,$_____$, 1/4$

Identify the pattern and fill up the number in the blank.

$-3, -5, -7,$_____, $-11$

Consider the following sequences

$1, \frac{-1}{2} ,\frac{1}{4} ,\frac{-1}{8} ,$_____$,\frac{-1}{32}$

$2, 3, 5, 7, 11, 13,$_____

$1, 4, 9, 16,$_____

$15, 12, 9, 6, 3, 0, -3, -6,$ _____