Shown below is a house of cards, a structure created by stacking playing cards on top of each other in the shape of pyramid. Each small triangle is made of $3$ cards and each layer has $1$ less triangle than the layer below it.

Ankit and his friends were having a sleepover and wanted to do something fun. One of the friends suggested they could make a house of cards.

Ankit is planning to make a pyramid with the top and bottom layer containing $15$ and $138$ cards respectively.

How many layers will such a pyramid have? Show your work.

Shown below is a house of cards, a structure created by stacking playing cards on top of each other in the shape of pyramid. Each small triangle is made of $3$ cards and each layer has $1$ less triangle than the layer below it.

Ankit and his friends were having a sleepover and wanted to do something fun. One of the friends suggested they could make a house of cards.

Ankit and his friends want to use $3$ cards in the top layer and $18$ in the bottom layer.

Form an AP showing the number of cards in each layer starting from the top layer.

The average of an Arithmetic progression with $151$ terms is zero. One of its terms is zero.

Which term of the Arithmetic Progression is zero? Show your steps.

Huner said, "The value of the $20$th term of ANY arithmetic progression is double that of the $10$th term."

Is Huner's statement correct? Justify your answer.

Shivam bought a large quantity of marigold flowers to decorate his house for a family function. He used $630$ flowers to recreate the pattern shown below. He used $7$ flowers in the shortest garland and $35$ flowers in the longest garland.

If he kept the difference between two consecutive garlands the same, how many garlands did he make? Show your steps.

Parth was receiving spam calls from a telemarketing centre. He got the first call at $2:48 \mathrm{pm}$ as shown below.

If the telemarketer continues to call using the same pattern, at what time will Parth receive the $13$th call? Show your work.

Ramit is moving towards a stationary object of sound at a constant speed. At his initial position, he hears a pitch of $5$ hertz.

If the pitch from the source increases by $4$ hertz for every $5$ m Ramit travels, find the distance he would have travelled towards the source when the pitch is exactly $149$ hertz. Show your steps.

A stone is thrown into still water and the figure below represents the concentric circular phenomenon known as ripple effect. The radius of first circle is $3$ cm.

If the radius of each subsequent circle is $4$ cm more than the previous, which circle has a radius of $43$ cm? Show your steps.

$14,21,28,25,...$and $26,39,52,65,..$are two arithmetic progressions such that the $p$th term of the first arithmetic progression is the same as the $q$th term of the second arithmetic progression.

Derive a relationship between $p$ and $q$. Show your work.