Animation is a method in which a sequence of images are manipulated to appear as moving objects. An animation specialist wants to show the growth of a sapling into a tree through animation. She follows the steps below:

• She develops the first image by designing a sapling containing a certain number of leaves.
• She develops the second image by adding $15$ leaves to the first image.
• She develops the third image by adding $22$ leaves to the second image.
• She develops the fourth image by adding $29$ leaves to the third image and so on.

If she continues the process in the same manner, how many leaves will she be adding to the $25$th image to develop $26$th image? Show your work.

The sum of first two terms of an arithmetic progression is the same as the sum of the first seven terms of the same arithmetic progression.

Can such an arithmetic progression exist? Justify your answer,

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?