Question

In the sequence 1, 3, 5, 7, ..., k, ..., 57, the sum of the numbers up to k, excluding k, is equal to the sum of the numbers from k up to 57, also excluding k. What is k?

Hint:

The sum of the first n odd numbers is \(n^2\). Remembering this shortcut is much faster than using the general AP sum formula for this type of problem.

Answer 1

Correct Answer: 41

Step 1: Describe the given sequence. The sequence consists of consecutive odd numbers: \[ 1, 3, 5, \ldots, 57. \] This is an arithmetic progression with general term \[ a_n = 2n - 1. \] Step 2: Find the total number of terms. Let the total number of terms be \(N\). Since the last term is 57, \[ 2N - 1 = 57 \Rightarrow 2N = 58 \Rightarrow N = 29. \] Hence, the sequence has 29 terms. The sum of all terms is \[ S_{29} = 29^2 = 841. \] Step 3: Represent the position of \(k\). Let \(k\) be the \(m\)-th term of the sequence. Then, \[ k = a_m = 2m - 1. \] The sum of all terms before \(k\) is the sum of the first \(m-1\) odd numbers: \[ S_{\text{before}} = (m-1)^2. \] The sum of all terms after \(k\) is the total sum minus the sum of the first \(m\) terms: \[ S_{\text{after}} = 29^2 - m^2. \] Step 4: Use the given condition. It is given that the sum of terms before \(k\) equals the sum of terms after \(k\): \[ (m-1)^2 = 29^2 - m^2. \] Expanding and simplifying, \[ m^2 - 2m + 1 = 841 - m^2, \] \[ 2m^2 - 2m - 840 = 0. \] Dividing by 2, \[ m^2 - m - 420 = 0. \] Factoring, \[ (m - 21)(m + 20) = 0. \] Thus, \[ m = 21 \quad \text{or} \quad m = -20. \] Since \(m\) must be positive, we take \(m = 21\). Step 5: Find the value of \(k\). \[ k = a_{21} = 2(21) - 1 = 41. \] Therefore, the value of \(k\) is \(41\).

Answer 2

Step 1: Understanding the Question:
We are given an arithmetic progression (AP) of odd numbers from 1 to 57.
There is a term 'k' in this sequence.
The problem states that the sum of all terms before 'k' is equal to the sum of all terms after 'k'.
Our goal is to find the value of 'k'.
Step 2: Key Formula or Approach:
The given sequence is an AP: 1, 3, 5, ...
The n-th term of this AP is given by \(a_n = 2n - 1\).
The sum of the first 'n' odd numbers is a standard result given by the formula:
\[ S_n = n^2 \] We will use this formula to represent the sums and solve for 'k'.
Step 3: Detailed Explanation:
First, let's find the total number of terms in the sequence. The last term is 57.
Let N be the total number of terms.
\[ a_N = 2N - 1 = 57 \] \[ 2N = 58 \] \[ N = 29 \] So, there are 29 terms in the entire sequence. The sum of all terms is \(S_{29} = 29^2 = 841\).
Now, let 'k' be the m-th term in the sequence. So, \(k = a_m = 2m - 1\).
The sum of terms before k is the sum of the first \(m-1\) terms.
\[ S_{before} = S_{m-1} = (m-1)^2 \] The sum of terms after k is the sum of terms from the (m+1)-th term to the 29th term.
This can be calculated as the total sum minus the sum of terms up to and including k.
The sum up to and including k is the sum of the first 'm' terms, \(S_m = m^2\).
\[ S_{after} = S_{total} - S_m = 29^2 - m^2 \] According to the problem statement, \(S_{before} = S_{after}\).
\[ (m-1)^2 = 29^2 - m^2 \] Now, we solve this equation for m:
\[ m^2 - 2m + 1 = 841 - m^2 \] \[ 2m^2 - 2m - 840 = 0 \] Dividing the equation by 2, we get:
\[ m^2 - m - 420 = 0 \] This is a quadratic equation. We can solve it by factoring. We need two numbers that multiply to -420 and add to -1. These numbers are -21 and 20.
\[ (m - 21)(m + 20) = 0 \] The possible values for m are \(m = 21\) or \(m = -20\). Since m represents the position of a term in a sequence, it must be a positive integer. Thus, we take \(m = 21\).
The question asks for the value of k, which is the 21st term.
\[ k = a_{21} = 2(21) - 1 \] \[ k = 42 - 1 = 41 \] Step 4: Final Answer:
The value of k is 41.