Problem 56 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Sum of certain consecutive odd positive integers is \(57^{2}-13^{2}\). Find them.

Step-by-Step Solution

Answer

The consecutive odd positive integers with a sum equal to \(57^2 - 13^2\) are: \[3729, 3731, 3733, ..., 3845\].

Step 1: Find the actual value of the difference

First, we have to find the actual value of the difference given, which is \(57^2 - 13^2\). The formula for the difference of two squares is: \[A^2 - B^2 = (A+B)(A-B)\] Using this formula with \(A = 57\) and \(B = 13\): \[57^2 - 13^2 = (57+13)(57-13)\]

Step 2: Calculate the values in the brackets

Now, we will find the values of \(57 + 13\) and \(57 - 13\): \(57 + 13 = 70\) \(57 - 13 = 44\)

Step 3: Multiply the values to get the total sum

Now, multiply the values obtained in the previous step: \(70 * 44 = 3080\) Now we know the sum of the consecutive odd positive integers is \(3080\).

Step 4: Find the series of consecutive odd positive integers

The sum of the first n odd positive integers is given by the formula: \[S = n^2\] So, we have to find an odd number which when subtracted from this sum, leaves us with a perfect square. Now, let's try to find such an odd number that when subtracted from \(3080\), it leaves a perfect square: For example, \(3081 - 53 = 3027\) which is not a perfect square. \(3081 - 15 = 3066\) which is not a perfect square. \(3081 - 1 = 3080\) which is not a perfect square. \(3081 - 21 = 3060 \) which is also not a perfect square. However, \(3080 - 45 = 3035\) The result, \(3035\), is not a perfect square, but trying to represent it as the difference of consecutive squares, \(55^2 - 10^2 = 3025\) Then: \(3080 - 45 = 3025 + 5\) So, the sum of consecutive odd positive integers is equal to \(3080\). Thus, the consecutive odd positive integers are: \(3025\), \(3027\), \(3029\), ..., \(3045\) Let's break this sum down into the series: \[5^2 + 7^2 + 11^2 + ... + 45\] \[5^2 + (5^2 + 2^2) + (5^2 + 6^2) + ... + 45\] Now, since there are 5 parts in this sum, we can represent these parts as following: \[5(5^2) + (0^2 + 2^2 + 6^2 + 8^2 + 12^2)\] \[5(25) + (0 + 4 + 36 + 64 + 144)\] \[125 + 248 = 373\] Now we have a sum of 373 with the consecutive odd positive integers starting at 3729 and ending at 3845. Thus, the consecutive odd positive integers with a sum equal to \(57^2 - 13^2\) are: \[3729, 3731, 3733, ..., 3845\]

Key Concepts

Difference of Squares

The difference of squares is a fascinating algebraic identity that states: \(A^2 - B^2 = (A+B)(A-B)\). This formula allows us to express the difference of two squared numbers as a product of a sum and a difference, which simplifies calculations.

• It turns a seemingly complex subtraction into a straightforward multiplication.
• For instance, finding \(57^2 - 13^2\) becomes easier when we calculate \((57+13)(57-13)\).

In our exercise, this identity simplifies our work. By applying it, the problem of subtracting two large squares becomes a manageable multiplication problem. Understanding this simplifies not only the calculation but also strengthens your problem-solving skills in algebra.

Sum of Integers

Finding the sum of integers, especially consecutive odd integers, involves specific techniques. An important formula here is \(S = n^2\), where \(S\) is the sum of the first \(n\) odd integers. This formula becomes instrumental when dealing with problems involving consecutive odd numbers.

• Odd integers follow a specific pattern: 1, 3, 5, 7, and so on.
• Each odd integer increases by 2 from the previous one.

In our exercise, knowing that the sum of some consecutive odd integers equals \(3080\) guides us to identify which integers could fit. By representing the sum as a difference, we ensure that it aligns perfectly with our calculations using squares and the identity of consecutive integers.

Algebra

Algebra is like a puzzle where numbers and letters mix to solve problems. It involves using equations to express relationships between different variables.

• It allows for unknown numbers (like \(x\) or \(n\)) to be solved through equations.
• Algebraic techniques help manipulate expressions and reveal new insights into problems.

In this exercise, algebra helps us decipher the relationships between consecutive odd integers and their sums. By setting up an equation involving the known sum \(3080\) and unknown consecutive integers, we use algebraic manipulation to isolate and identify these integers. This approach is essential for bridging the gap between numerical problems and abstract solutions.

Perfect Squares

Perfect squares are numbers that give whole numbers when square-rooted. They are products of an integer multiplied by itself, such as \(1 \times 1 = 1\), \(2 \times 2 = 4\), etc.

• They form the foundation for many mathematical concepts, including the difference of squares.
• Every positive integer has a unique perfect square closest to it.

In our problem, identifying perfect squares helps narrow down which integers add up to \(3080\). Since \(3080\) is broken down into parts reflecting such perfect squares like \(57^2 - 13^2\), we could determine the series of integers involved. Recognizing these patterns simplifies solving for consecutive integers by leveraging properties of perfect squares.