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Observe the Following Pattern : 13 = 1 13 + 23 = (1 + 2)2 13 + 23 + 33 = (1 + 2 + 3)2 Write the Next Three Rows and Calculate the Value of 13 + 23 + 33 + ...+ 93 + 103 by the Above Pattern. - Mathematics

Sum

Observe the following pattern:
                13 = 1
        13 + 23 = (1 + 2)2
13 + 23 + 33 = (1 + 2 + 3)2
Write the next three rows and calculate the value of 13 + 2+ 33 + ... + 9+ 103 by the above pattern.

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Solution

Extend the pattern as follows:

\[1^3 = 1\]

\[ 1^3 + 2^3 = \left( 1 + 2 \right)^2 \]

\[ 1^3 + 2^3 + 3^3 = \left( 1 + 2 + 3 \right)^2 \]

\[ 1^3 + 2^3 + 3^3 + 4^3 = \left( 1 + 2 + 3 + 4 \right)^2 \]

\[ 1^3 + 2^3 + 3^3 + 4^3 + 5^3 = \left( 1 + 2 + 3 + 4 + 5 \right)^2 \]

\[ 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 = \left( 1 + 2 + 3 + 4 + 5 + 6 \right)^2 \]

Now, from the above pattern, the required value is given by:

\[1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 + {10}^3 = \left( 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 \right)^2 = {55}^2 = 3025\]

Thus, the required value is 3025.

 
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APPEARS IN

RD Sharma Class 8 Maths
Chapter 4 Cubes and Cube Roots
Exercise 4.1 | Q 3 | Page 8
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