Question

If \(a_{1}, a_{2}, a_{3}, \ldots\) is an A.P. such that \(a_{1}+a_{5}+a_{10}+a_{15}+a_{20}+a_{24}=225\) then \(a_{1}+a_{2}+a_{3}+\ldots+a_{23}+a_{24}\) is equal to : (a) 999 (b) 900 (c) 1225 (d) none of these

Short answer

Given that: - The sum of the terms: \(a_{1}+a_{5}+a_{10}+a_{15}+a_{20}+a_{24}=225\) - Find: \(a_1 + a_2 + \ldots + a_{23}+ a_{24}\) Answer: Based on our analysis and calculations, the sum of the first 24 terms of the arithmetic progression is equal to 900.

Step-by-step solution

Express the given terms in terms of the first term and the common difference

Use the arithmetic progression definition to express the terms \(a_5, a_{10}, a_{15}, a_{20}, a_{24}\) in terms of \(a_1\) and the common difference \(d\): $$a_{n}=a_1+(n-1)d$$ \(a_{5}=a_1+(5-1)d=a_1+4d\) \(a_{10}=a_1+(10-1)d=a_1+9d\) \(a_{15}=a_1+(15-1)d=a_1+14d\) \(a_{20}=a_1+(20-1)d=a_1+19d\) \(a_{24}=a_1+(24-1)d=a_1+23d\)

Create an equation using the given sum

Plug in the expressions from step 1 into the given sum: $$a_{1}+a_{5}+a_{10}+a_{15}+a_{20}+a_{24}=225$$ Substitute the expressions: $$a_1+(a_1+4d)+(a_1+9d)+(a_1+14d)+(a_1+19d)+(a_1+23d)=225$$

Solve for one of the variables

Simplify and solve for one of the variables, let's say \(a_1\): $$6a_1+69d=225$$ Now, let's express \(a_1\) in terms of \(d\): $$a_1=\frac{225-69d}{6}$$

Find the sum of the first 24 terms

Now, we need to find the sum of the first 24 terms of the arithmetic progression. The sum of an arithmetic progression with \(n\) terms is given by: $$S_n=\frac{n}{2}(2a_1+(n-1)d)$$ We are looking for \(S_{24}\), so we plug in \(n=24\) : $$S_{24}=\frac{24}{2}(2a_1+(24-1)d)$$ Substitute the expression for \(a_1\) from Step 3: $$S_{24}=\frac{24}{2}(2\frac{225-69d}{6}+(23)d)$$

Simplify and evaluate the sum

Simplify the expression and evaluate the sum: $$S_{24}=12(\frac{(225-69d)}{3}+(23)d)$$ $$S_{24}=4(225-69d+69d)=900$$ So, the sum of the first 24 terms \(a_1+a_2+\ldots+a_{23}+a_{24}\) is equal to \(\boxed{900}\), which means the correct answer is (b) 900.

Key concepts

Sum of arithmetic series

The sum of an arithmetic series is a useful concept when dealing with sequences of numbers that have a common difference between consecutive terms. To find this sum, we use the formula:\[ S_n = \frac{n}{2} (2a_1 + (n-1)d) \]where:

• \(S_n\) is the sum of the first \(n\) terms.
• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference between the terms.
• \(n\) is the number of terms to add together.

In our exercise, we found the sum \(S_{24}\) by substituting the known values into this formula. We calculated the terms using a given equation and the first term expression to eventually find that the sum of the first 24 terms was 900. Remember, the key is to apply the formula correctly, ensuring each element like common difference and first term is perfectly determined for accurate results.

Common difference in AP

In an arithmetic progression, every consecutive pair of terms has a fixed amount added to the previous term, known as the common difference (\(d\)). Consequently, if you know one term and its position in the sequence, you can find any other term using:\[ a_n = a_1 + (n-1)d \]Here, for instance, by expressing terms like \(a_5\), \(a_{10}\), and others as equations in terms of \(a_1\) and \(d\), we effectively harnessed the definition of arithmetic progressions. The task was to find a particular sum by first understanding how each term builds upon the last. The sequence's consistency (determined by \(d\)) is what makes these calculations achievable and predictable.

First term in AP

The first term of an arithmetic progression (denoted \(a_1\)) is the starting point of the series. It's crucial because every subsequent term is derived starting from this point, progressing by adding the common difference. In the steps presented, the first term \(a_1\) was expressed in terms of the common difference by manipulating the given sum formula: \[ 6a_1 + 69d = 225 \]From this pivotal equation, we obtained \(a_1\) as:\[ a_1 = \frac{225 - 69d}{6} \]Therefore, understanding how to express \(a_1\) with respect to \(d\) is essential in answering problems related to arithmetic progressions. It forms the groundwork from which we apply further formulas, such as those for the sum of the series.