Find the sum of first 24 terms of the A.P. $a_{1}, a_{2}, a_{3}, \ldots$ if it is known that $a_{1}+a_{5}+a_{10}+a_{15}+a_{20}+a_{24}=225$.

Step 1: Understand the properties of an A.P.

In an Arithmetic Progression (A.P.), the $n$ -th term is given by the formula $a_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference. A key property of an A.P. is that the sum of terms equidistant from the beginning and end is constant. For an A.P. with $N$ terms, $a_k + a_{N-k+1}$ is constant.

Step 2: Apply the property of equidistant terms

For an A.P. with 24 terms, the sum of terms equidistant from the beginning and end is equal. Here, $a_1$ and $a_{24}$ are the first and last terms. $a_5$ is the 5th term and $a_{20}$ is the 5th term from the end ($24 - 5 + 1 = 20$). Similarly, $a_{10}$ is the 10th term and $a_{15}$ is the 10th term from the end ($24 - 10 + 1 = 15$). Therefore, $a_1 + a_{24} = a_5 + a_{20} = a_{10} + a_{15}$.

Step 3: Substitute into the given equation

We are given the equation $a_1+a_5+a_{10}+a_{15}+a_{20}+a_{24}=225$. Using the property from the previous step, we can group these terms: $(a_1 + a_{24}) + (a_5 + a_{20}) + (a_{10} + a_{15}) = 225$. Since each of these pairs sums to the same value, let's call it $S'$, we have $S' + S' + S' = 225$, or $3S' = 225$. This gives us $S' = a_1 + a_{24} = 75$.

Step 4: Calculate the sum of the first 24 terms

The sum of the first $n$ terms of an A.P. is given by the formula $S_n = \frac{n}{2}(a_1 + a_n)$. For the first 24 terms, $n=24$. We have already found that $a_1 + a_{24} = 75$. Substituting these values into the formula, we get $S_{24} = \frac{24}{2}(75) = 12 \times 75 = 900$.

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