If $\frac{3 + 5 + 7 + \dots + (2n + 1)}{5 + 8 + 11 + \dots + (3n + 2)} = 7$,find the value of $n$. (Note: The original problem implies $n$ terms in the numerator and $10$ terms in the denominator as per the provided solution structure).

If the sum of three numbers in an arithmetic sequence is $15$ and the sum of their squares is $83$,then the numbers are

$A$ person is to count $4500$ currency notes. Let $a_n$ denote the number of notes he counts in the $n^{th}$ minute. If $a_1 = a_2 = \ldots = a_{10} = 150$ and $a_{10}, a_{11}, \ldots$ are in an $A.P.$ with common difference $-2$,then the time taken by him to count all notes is ............... $minutes$.

If the average of the first $n$ numbers in the sequence $148, 146, 144, \ldots$ is $125$,then $n$ is equal to

The sum of the first $p, q,$ and $r$ terms of an $A.P.$ are $a, b,$ and $c,$ respectively. Prove that $\frac{a}{p}(q-r)+\frac{b}{q}(r-p)+\frac{c}{r}(p-q)=0$.

If ${a_1}, {a_2}, {a_3}, \dots, {a_{24}}$ are in arithmetic progression and ${a_1} + {a_5} + {a_{10}} + {a_{15}} + {a_{20}} + {a_{24}} = 225$,then ${a_1} + {a_2} + {a_3} + \dots + {a_{23}} + {a_{24}} = $