The houses of a row are numbered consecutively from $1$ to $49$. Show that there is a value of $x$ such that the sum of the numbers of the houses preceding the house numbered $x$ is equal to the sum of the numbers of the houses following it. Find this value of $x$.

$A$ small terrace at a football ground comprises of $15$ steps each of which is $50 \, m$ long and built of solid concrete. Each step has a rise of $\frac{1}{4} \, m$ and a tread of $\frac{1}{2} \, m$. Calculate the total volume of concrete required to build the terrace (in $m^3$).

In which of the following situations,does the list of numbers involved make an arithmetic progression,and why? The amount of money in the account every year,when ₹ $10000$ is deposited at compound interest at $8 \%$ per annum.

Are $2, 4, 8, 16, \ldots$ in an $AP$? If they form an $AP$,find the common difference $d$ and write three more terms.

$A$ spiral is made up of successive semicircles,with centres alternately at $A$ and $B$,starting with centre at $A$,of radii $0.5 \, cm, 1.0 \, cm, 1.5 \, cm, 2.0 \, cm, \ldots$ as shown in the figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take $\pi = \frac{22}{7}$)

Check whether $-150$ is a term of the $AP: 11, 8, 5, 2, \ldots$