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Question: The sum of the series \({1^3} - {2^3} + {3^3} - ..... + {9^3} = ?\) A) 300 B) 125 C) 425 D)...

The sum of the series 13−23+33−.....+93=?{1^3} - {2^3} + {3^3} - ..... + {9^3} = ?
A) 300
B) 125
C) 425
D) 0

Explanation

Solution

Here we need to find the sum of the given series. For that, first we will divide the series into two parts using mathematical operations i.e. addition and subtraction and these two parts will contain the sum of the cubes. Then we will use the formula for the sum of these cubes. From there, we will get the required sum of the series.

Complete step by step solution:
The given series is 13−23+33−.....+93{1^3} - {2^3} + {3^3} - ..... + {9^3}. Here we need to find the sum of this series.
Now, we will divide this series into parts using addition and subtraction in such a way that two part will contain the sum of the cubes.
=(13+23+33+.....+93)−2(23+43+63+83)= \left( {{1^3} + {2^3} + {3^3} + ..... + {9^3}} \right) - 2\left( {{2^3} + {4^3} + {6^3} + {8^3}} \right)
Now, we will take the common 23{2^3} from the second part.
=(13+23+33+.....+93)−2⋅23(13+23+33+43)= \left( {{1^3} + {2^3} + {3^3} + ..... + {9^3}} \right) - 2 \cdot {2^3}\left( {{1^3} + {2^3} + {3^3} + {4^3}} \right)
Now, we will use the formula of addition of cubes which is give as
(13+23+33+.....+n3)=n2(n+1)24\left( {{1^3} + {2^3} + {3^3} + ..... + {n^3}} \right) = \dfrac{{{n^2}{{\left( {n + 1} \right)}^2}}}{4}
Applying this formula in both parts of the series, we get
=92⋅(9+1)24−2⋅23⋅42⋅(4+1)24= \dfrac{{{9^2} \cdot {{\left( {9 + 1} \right)}^2}}}{4} - 2 \cdot {2^3} \cdot \dfrac{{{4^2} \cdot {{\left( {4 + 1} \right)}^2}}}{4}
Now, we will simplify the terms.
=452−24⋅100= {45^2} - {2^4} \cdot 100
On multiplying the terms and applying the exponents on the bases, we get
=2025−1600= 2025 - 1600
On subtracting these numbers, we get
=425= 425
Thus, the sum of the given series is equal to 425.

Hence, the correct option is option C.

Note:
Here we have calculated the sum of the given series using the formula of the series. But we can also calculate the cube of the numbers individually to reach our needed result when the series is short but we can’t do this when the series is too long. We always use the formula of the sum of cubes when the series is long. Remember to make the problem short and easy we need to remember the formula of sum of cubes.