Question

The sum of the first 51 terms of the arithmetic progression whose 2 nd term is 2 and 4 th term is 8 , (1) 3774 (2) 3477 (3) 7548 (4) 7458

Short answer

Answer: (1) 3774

Step-by-step solution

Understand the formula of arithmetic progression

In an arithmetic progression, the formula to find the nth term is given by: an = a + (n-1)d where an is the nth term, a is the first term, n is the position of the term, and d is the common difference.

Find the common difference

We are given that the 2nd term is 2 and the 4th term is 8. Let's apply the formula to both terms: a2 = a + d = 2 a4 = a + 3d = 8

Solve the system of equations

We can solve the system of equations by substitution or elimination. In this case, we'll use substitution: From the first equation, a = 2 - d. Substitute this into the second equation: (2 - d) + 3d = 8 Solve for d: 2d = 6 d = 3 Now, substitute the value of d back into the equation a = 2 - d: a = 2 - 3 a = -1

Find the sum of the first 51 terms

Now that we know the first term a and the common difference d, we can find the sum of the first 51 terms using the formula for the sum of an arithmetic progression: Sn = (n/2)(2a + (n-1)d) S51 = (51/2)(2(-1) + (51-1)3)

Calculate and find the correct answer

Plug in the values and compute the sum: S51 = (51/2)(-2 + 150) S51 = (51/2)(148) S51 = 3774 Comparing this sum to the given options, we can now conclude that the correct answer is (1) 3774.

Key concepts

Mathematical Induction

Understanding mathematical induction is akin to observing a row of dominoes topple one after the other. In mathematics, we use induction to prove that a statement is true for all natural numbers. It consists of two crucial steps: the base case, where we verify the statement for the initial value, often for the number 1; followed by the inductive step, where we assume the statement is true for some natural number 'k' and then prove it for 'k+1'. This process is akin to proving the first domino falls, and showing that if any one domino falls, then the next one will, ensuring the entire line of dominoes will inevitably fall.

For example, the formula for the sum of an arithmetic progression could be proven using mathematical induction by showing it holds true for the first term and then, assuming it works for 'k' terms, proving it must also work for 'k+1' terms. In the provided exercise, while not explicitly stated, understanding inductive reasoning helps grasp why the formulas used to find the nth term and the sum hold true across the board for any arithmetic progression.

Summation of Series

The summation of a series involves finding the total of all terms within the series. When we have a series formed by an arithmetic progression, the sum is calculated using a specific formula, which is remarkably useful for determining total values without needing to add each term individually. The formula for the sum of the first 'n' terms of an arithmetic progression is given by:
\[ S_n = \frac{n}{2}(2a + (n-1)d) \]
where 'n' denotes the number of terms, 'a' is the first term and 'd' is the common difference. In the provided exercise, we applied this formula to calculate the sum of the first 51 terms. Not only does this approach save time, but it also ensures accuracy by avoiding the tedious process of manual addition. It's important to overall understand how this formula derives from the average of the first and last term, multiplied by the number of terms, emphasizing the beauty and efficiency of arithmetic progression in mathematics.

Geometric Sequence

While not directly related to the exercise, understanding a geometric sequence can help differentiate it from an arithmetic progression. A geometric sequence refers to a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The formula to find the nth term (\( a_n \)) in a geometric sequence is \[ a_n = a_1 \times r^{(n-1)} \]
where \( a_1 \) is the first term, 'n' is the term number, and 'r' is the common ratio. Although the problem at hand involves an arithmetic progression where we add a constant to go to the next term, contrasting it with a geometric sequence, where we multiply by a constant, helps solidify one's understanding of sequences and series in math. This comparison serves to enhance comprehension of how different consistent patterns lead to differing types of sequences and their respective methods of analysis and summation.