The average of 101 consecutive odd numbers is 303. Find the largest number.
403
Solution:
Step 1: Understand the properties of consecutive odd numbers
For an odd number of consecutive odd numbers, the average equals the middle number.
Since we have 101 numbers (an odd count), the 51st number is the middle number.
Step 2: Identify the middle number
Given average = 303, which is the middle (51st) number:
Middle number = 303
Step 3: Determine the position of the largest number
In 101 consecutive odd numbers:
Largest number = Middle number + (2 × (50))
Because there are 50 numbers above the middle number, and each consecutive odd number increases by 2.
Step 4: Calculate the largest number
Largest number = 303 + (2 × 50) = 303 + 100 = 403
Conclusion:
The largest number in the sequence is 403.
Therefore, the correct option is 3.
403
Given:
The average of 101 consecutive odd numbers is 303.
Step 1: Use the average formula for consecutive odd numbers
For an odd sequence with an odd number of terms, the average is equal to the middle number.
Since the average is 303, the middle number is:
303
Step 2: Find the largest number in the sequence
The sequence contains 101 odd numbers, which means:
- The middle number is at position: (101 + 1) / 2 = 51st term
- The largest number is at the last position, which is the 101st term.
Use the formula for the last term:
Largest number = Middle number + (number of steps to the end × 2)
Largest number = 303 + (50 × 2)
Largest number = 303 + 100 = 403
✅ Conclusion:
The largest number in the sequence is: 403
403
Understanding Consecutive Odd Numbers:
- Consecutive odd numbers have a difference of 2 between them (e.g., 1, 3, 5, 7...).
The Average as the Middle Number:
- The average of an odd number of consecutive numbers is the middle number.
- Since there are 101 consecutive odd numbers, the middle number is the (101 + 1) / 2 = 51st number.
- We are given that the average is 303, so the 51st number is 303.
Finding the Largest Number:
- We want to find the 101st number.
- The difference between the 51st and 101st number is 101 - 51 = 50 intervals.
- Each interval is 2, so the total difference is 50 * 2 = 100.
- The largest number is 303 + 100 = 403.
Therefore, the largest number is 403.
Important Questions from Probability
A die is rolled thrice. What is the probability of getting a number greater than 4 in the first and the second throws, and a number less than 4 in the third throw?
Two dice are thrown simultaneously. If \( X \) denotes the number of fours, then the expectation of \( X \) will be:
If the random variable \( X \) has the following distribution:
X 0 1 2 otherwise P(X) k 2k 3k 0 Match List-I with List-II:
List-I List-II (A) k (I) \(\frac{5}{6}\) (B) P(X < 2) (II) \(\frac{4}{3}\) (C) E(X) (III) \(\frac{1}{2}\) (D) P(1 ≤ X ≤ 2) (IV) \(\frac{1}{6}\) Choose the correct answer from the options given below:
Let X denote the number of hours you play during a randomly selected day. The probability that X can take values x has the following form, where c is some constant:
\[ P(X = x) = \begin{cases} 0.1, & \text{if } x = 0 \\ cx, & \text{if } x = 1 \text{ or } x = 2 \\ c(5 - x), & \text{if } x = 3 \text{ or } x = 4 \\ 0, & \text{otherwise} \end{cases} \]
Match List-I with List-II:
List-I List-II (A) c (I) 0.75 (B) P(X ≤ 2) (II) 0.3 (C) P(X = 2) (III) 0.55 (D) P(X ≥ 2) (IV) 0.15 Choose the correct answer from the options given below:
For the differential equation \( (x \log_e x) dy = (\log_e x - y) dx \):
(A) Degree of the given differential equation is 1.
(B) It is a homogeneous differential equation.
(C) Solution is \( 2y \log_e x + A = (\log_e x)^2 \), where A is an arbitrary constant.
(D) Solution is \( 2y \log_e x + A = \log_e (\log_e x) \), where A is an arbitrary constant.
Choose the correct answer from the options given below:
