Find the sum of first $22$ terms of an $AP$ in which $d=7$ and $22^{nd}$ term is $149$.

Find the $31^{st}$ term of an $AP$ whose $11^{th}$ term is $38$ and the $16^{th}$ term is $73.$

Are $3, 3+\sqrt{2}, 3+2\sqrt{2}, 3+3\sqrt{2}, \ldots$ in $AP$? If they form an $AP$,find the common difference $d$ and write the next three terms.

The sum of the third and the seventh terms of an $AP$ is $6$ and their product is $8$. Find the sum of the first sixteen terms of the $AP$.

$A$ 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 $m^3$).

Does the sequence $-2, 2, -2, 2, -2, \ldots$ form an $AP$? If it forms an $AP$,write the next two terms.