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Chapter 5: Problem 85

\(1+4+9+16+25+\ldots \ldots \ldots+400=\) (a) 2780 (b) 2870 (c) 4280 (d) 2650

Short Answer

Expert verified
Answer: 2870

Step by step solution

01

Identify the series

We are given the series \(1^2+2^2+3^2+4^2+5^2+\ldots+20^2\). This is a sum of the first 20 perfect squares.
02

Apply the formula for the sum of the first n perfect squares

According to the formula, the sum of the first n perfect squares is given by: \(\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}\).
03

Calculate the sum of the first 20 perfect squares using the formula

Now, we will use the formula to calculate the sum of the first 20 perfect squares: \(\sum_{k=1}^{20} k^2 = \frac{20(20+1)(2(20)+1)}{6} =\frac{20(21)(41)}{6} =2870\).
04

Choose the correct answer from the options provided

We have found the sum of the given series to be 2870, which corresponds to the option (b). Thus, the correct answer is (b) 2870.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arithmetic Series
An arithmetic series is a sequence of numbers in which each term after the first is found by adding a constant, known as the common difference, to the previous term. The series looks like this: if the first term is 'a' and the common difference is 'd', then the series is:

\[a, a + d, a + 2d, a + 3d, \text{...} \text{,} a + (n-1)d\].

However, the series given in the exercise is not an arithmetic series because the difference between consecutive terms grows larger with each step. It represents a series of perfect squares, each number being the square of its position in the sequence: \(1^2, 2^2, 3^2, \text{...}, n^2\). Even though this is not an arithmetic series, understanding arithmetic series lays the groundwork for recognizing different types of sequences and series in mathematics.
Mathematical Induction
Mathematical induction is a powerful proof technique used in mathematics to establish that a given statement is true for all natural numbers. It consists of two steps:

  • Base Case: Prove the statement is true for the initial value, usually 1.
  • Inductive Step: Assume the statement is true for a certain natural number 'k', and then prove it is true for 'k+1'.

This technique can be applied to prove the correctness of formulas, such as the formula for the sum of the first 'n' perfect squares. Induction helps verify the validity of the formula step by step, thereby providing assurance that it will work for any natural number 'n'.
Algebraic Formulas
Algebraic formulas are equations that describe relationships among variables and constants. They are tools that allow us to solve problems across various fields of mathematics, including geometry and algebra. The formula used in the solution for the sum of the first 'n' perfect squares,

\[\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}\],

is a well-known algebraic formula. It is derived by mathematical induction and provides a shortcut to quickly find the sum of squares without needing to add up each term individually. Such formulas are essential for students to remember and understand as they significantly reduce the complexity of problem-solving in mathematics.
IIT-JEE Mathematics
The Indian Institute of Technology Joint Entrance Examination (IIT-JEE) is a highly competitive exam that assesses a student's understanding of core concepts in physics, chemistry, and mathematics. For mathematics, the IIT-JEE covers a wide range of topics including algebra, calculus, trigonometry, and more. A strong grasp of mathematical formulas and the ability to apply them to solve complex problems is essential. The exercise given here is an example of the type of problem that could appear in the IIT-JEE. These problems not only test students' knowledge of formulas but also their ability to recognize patterns, conceptual understanding, and problem-solving skills. Preparing for such problems often involves practicing with textbook exercises and understanding the derivation and application of various mathematical formulas.

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Most popular questions from this chapter

Column I \(\quad\) Column II (a) If \(\mathrm{a}^{x}=\mathrm{b}^{\gamma}=\mathrm{c}^{x}\) where \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are in \(\mathrm{GP}\), then \(\mathrm{x}, y, \mathrm{z}\) are in (p) AP (b) Three distinct numbers \(a, b, c\) satisfying \(\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}\) are in (q) \(\mathrm{GP}\) (c) If \(\mathrm{x}, \mathrm{y}, \mathrm{z}(\mathrm{all}>1)\) are in GP, then \(\frac{1}{1+\log \mathrm{x}}, \frac{1}{1+\log \mathrm{y}}, \frac{1}{1+\log \mathrm{z}}\) are in (r) HP (d) Three consecutive terms \(\frac{1}{1+\sqrt{x}}, \frac{1}{1-x}, \frac{1}{1-\sqrt{x}}\) of a sequence are in (s) AGP

If the second number of three numbers in an increasing \(\mathrm{GP}\) is doubled, we get an \(\mathrm{AP}\), then (a) common ratio of the GP is 1 or \(-1\) (b) common ratio of the GP and common difference of AP are equal. (c) common difference of AP is equal to both first term and last term. (d) common ratio of the GP is \(2+\sqrt{3}\).

If the ratio of the sum of the first \(\mathrm{m}\) terms and the first \(\mathrm{n}\) terms of an \(\mathrm{AP}\) is \(\mathrm{m}^{2}: \mathrm{n}^{2}\), the ratio of its \(\mathrm{mth}\) and \(\mathrm{nth}\) term will be (a) \(2 \mathrm{~m}-1: 2 \mathrm{n}-1\) (b) \(\mathrm{m}: \mathrm{n}\) (c) \(2 \mathrm{~m}+1: 2 \mathrm{n}+1\) (d) None of these

If \(a x^{2}+2 b x+c=0\) and \(a_{1} x^{2}+2 b_{1} x+c_{1}=0\) have a common root, and if \(\frac{a}{a_{1}}, \frac{b}{b_{1}}, \frac{c}{c_{1}}\) are in AP then \(a_{1}, b_{1}, c_{1}\) are in (a) GP (b) AP (c) HP (d) AGP

If \(\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\) are in GP then (a) \(\mathrm{a}+\mathrm{b}, \mathrm{b}+c_{2} \mathrm{c}+\mathrm{d}\) are in \(\mathrm{GP}\) (b) \(a x^{2}+c\) is a factor of \(a x^{3}+b x^{2}+c x+d\) (c) \(a^{2}+b^{2}+c^{2}, a b+b c+c d, b^{2}+c^{2}+d^{2}\) are in GP (d) \(a x+c\) is a factor of \(a x^{3}+b x^{2}+c x+d\)
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