Question

For a sequence of real numbers \(x_1, x_2, ..., x_n,\) if \(x_1 - x_2 + x_3 - ... + (-1)^{n + 1}x_n =n^2 + 2n\) for all natural numbers n, then the sum \(x_{49} + x_{50}\) equals

• A. 2
• B. -2
• C. 200
• D. -200

Answer 1

The Correct Option is B

We are given a sequence where the first term is:  

Next, we are told that: 
\(x_1 - x_2 = 8\) 
Substituting \(x_1 = 3\), we get: 
\(3 - x_2 = 8 \Rightarrow x_2 = -5\)

Now using the next part of the series: 
\(x_1 - x_2 + x_3 = 15\) 
Substitute \(x_1 = 3\) and \(x_2 = -5\): 
\(3 - (-5) + x_3 = 15 \Rightarrow 8 + x_3 = 15 \Rightarrow x_3 = 7\)

Now we try to identify the pattern of the sequence: 
Observing the values of the terms: 
\(x_1 = 3 = (+1)(2×1+1)\)
\(x_2 = -5 = (-1)(2×2+1)\)
\(x_3 = 7 = (+1)(2×3+1)\)

From this, we see that the general term of the sequence can be written as: 
\(x_n = (-1)^{n+1}(2n+1)\)

Now, we calculate: 
\(x_{49} = (-1)^{50}(2×49+1) = (-1)^{50}(99) = +99\) 
\(x_{50} = (-1)^{51}(2×50+1) = (-1)^{51}(101) = -101\)

Therefore, the sum of these two terms is: 
\(x_{49} + x_{50} = 99 + (-101) = -2\)

Answer 2

Given: The sum of the first n terms of a series is expressed by the formula: 

\(S_n = x_1 - x_2 + x_3 - x_4 + \ldots + (-1)^{n+1} x_n = n^2 + 2n = n(n + 2)\)

This implies that the signs of the terms alternate: odd-positioned terms (e.g., \(x_1, x_3, x_5, \ldots\)) are positive, and even-positioned terms (e.g., \(x_2, x_4, x_6, \ldots\)) are negative.

To determine \(x_{49}\) and \(x_{50}\), we first calculate the values of:

• \(S_{50} = 50 \times (50 + 2) = 50 \times 52 = 2600\)
• \(S_{49} = 49 \times (49 + 2) = 49 \times 51 = 2499\)
• \(S_{48} = 48 \times (48 + 2) = 48 \times 50 = 2400\)

 

From the recursive definition of the series: 
To find \(x_{50}\):
\(S_{50} = S_{49} - x_{50} \Rightarrow x_{50} = S_{49} - S_{50} = 2499 - 2600 = -101\)

To find \(x_{49}\):
\(S_{49} = S_{48} + x_{49} \Rightarrow x_{49} = S_{49} - S_{48} = 2499 - 2400 = 99\)

Therefore, the sum of the 49th and 50th terms is:
\(x_{49} + x_{50} = 99 + (-101) = -2\)

Final Answer: \(-2\). Hence, the correct option is (B).