Problem 25 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the sum of all 2 digit odd numbers.

Step-by-Step Solution

Answer

The sum of all 2-digit odd numbers is 2475.

Step 1: Identifying the Range

For this exercise, we want to find the sum of all 2-digit odd numbers. The first 2-digit odd number is 11, and the last one is 99. Now, we've got the range of our sequence.

Step 2: Identifying the Common Difference

We will be adding only odd numbers in the sequence, so the common difference between the terms will be 2. For example, 11 + 2 = 13, 13 + 2 = 15, and so on.

Step 3: Finding the Number of Terms

To find the number of terms in the sequence, we will use the arithmetic sequence formula: nth term = a + (n-1)d, where "a" is the first term, "n" is the term number, and "d" is the common difference. Since the nth term is 99, we can rewrite the formula as: 99 = 11 + (n-1)2 Now, we will solve for n: \(99 - 11 = 88\) \(88 = (n - 1) * 2\) Now, divide both sides by 2: \(\frac{88}{2} = n - 1\) \(44 = n - 1\) Now, add 1 to both sides: \(n = 44 + 1\) \(n = 45\) So, there are 45 terms in this sequence.

Step 4: Applying Arithmetic Series Formula

Now that we have the number of terms and common difference, we can apply the arithmetic series formula to find the sum: Sum = n * (a + end) / 2, where "n" is the number of terms, "a" is the first term, and "end" is the last term in the sequence. Sum = 45 * (11 + 99) / 2 Now, solve the equation: Sum = 45 * (110) / 2 Sum = 45 * 55 Sum = 2475 Therefore, the sum of all 2-digit odd numbers is 2475.

Key Concepts

Sum of Arithmetic Series

In arithmetic sequences, calculating the sum of the series is a common task. An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant is called the "common difference." To find the sum of an arithmetic series, we use the formula:

• Sum = \( n \times \frac{(a + l)}{2} \)

where:

• \( n \) is the number of terms in the sequence,
• \( a \) is the first term of the sequence,
• \( l \) is the last term of the sequence.

Using this approach streamlines the addition process, especially when dealing with large sets of numbers. For an arithmetic series, once you've identified the first and last terms as well as the number of terms, this formula gives you the total quickly and efficiently.

Odd Numbers

Odd numbers are integers that cannot be exactly divided by 2. When you divide an odd number by 2, you'll get a remainder of 1. In arithmetic sequences that only include odd numbers, the pattern follows a specific order. For example, if you start with 11, adding 2 gets you the next odd number, which is 13. This forms an arithmetic sequence because each term progresses by the same common difference, which in this case is 2. Odd numbers frequently appear in math problems and understanding their properties simplifies many calculations and concepts.

Two-Digit Numbers

Two-digit numbers range from 10 to 99. When focusing on specific subsets of these numbers, like odd numbers, we limit the range further. For example, in the problem of summing all two-digit odd numbers, we begin at 11 and end at 99. Two-digit numbers are pivotal in arithmetic exercises because they provide a manageable range that incorporates fundamental number properties. Working within this confine helps students grasp broader mathematical concepts, laying the groundwork for more advanced topics. This specific range allows for a focus on number patterns and arithmetic skills without overwhelming complexity.