Two arithmetic progressions have the same first term. The common difference of one progression is $4$ more than the other progression. $124$th term of the first arithmetic progression is same as $42$nd term of the second.

Find out set of possible values of the common differences. Show your work.

Sana decided to start practising for an upcoming marathon. She decided to gradually increase the duration. She ran for $10$ mins on day $1$ and increased the duration by $5$ minutes every day.

From which day onwards will she be running $2.5$ hours or more? Show your work.

Consider the list of numbers below.

$\frac{1}{2}+k, \frac{2}{3}+k, \frac{3}{4}+k, \frac{4}{5}+k,....$ where $k$ is an integer.

Is the above list of numbers an arithmetic progression? Justify your answer.

The exterior angles marked in each of the polygons below are in arithmetic progression.

Minal drew one such polygon with $n$ sides. The smallest exterior angle is $8°$ and each subsequent angle is $4°$ more than previous angle.

Find the number of sides of the polygon that Minal had drawn. Show your steps.

$\sqrt{2}, \sqrt{18}, \sqrt{50}, \sqrt{98}....$

Is the above pattern in AP? Justify your answer.

$4$ groups in a class were asked to come up with an arithmetic progression (AP). Shown below are their responses:

$\begin{array}{|cc|}\hline \mathrm{Group}& \mathrm{Arithmetic} \mathrm{Progression}\\ \mathrm{M}& 4,2,0,-2,....\\ \mathrm{N}& 41,38.5,36,33.5,...\\ \mathrm{O}& -19,-21,-23,-25,...\\ \mathrm{P}& -3,-3,-3,-3,...\\ \hline\end{array}$

Which of these groups correctly came up with an AP?

The cylindrical bumps on top of lego blocks are called studs.

Pragun has built a solid inverted lego pyramid as shown below. The number of studs in successive floors forms an arithmetic progression. Pragun figures out that the sum of the number of studs used in the first $p$ floors is given by $\left(6p^2-2p\right)$.

How many studs are there in the $5$th floor?