Question

Give answer with full explanation

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Answer 1

Answer :-

→ 42,640.

Step-by-step explanation :-

To find :-

$\tt \rightarrow {(10)}^{2} + {(11)}^{2} + {(12)}^{2} + ...... + {(50)}^{2} .$

We have,

$\sf \rightarrow {1}^{2} + {2}^{2} = \frac{2 \times 3 \times 5}{6} . \\ \\ \sf \rightarrow {1}^{2} + {2}^{2} + {3}^{2} = \frac{3 \times 4 \times 7}{6} . \\ \\ \sf \rightarrow {1}^{2} + {2}^{2} + {3}^{2} + {4}^{2} = \frac{4 \times 5 \times 9}{6} .$

Let last squared number of LHS be a .

The system followed in this equation is :

$\sf \implies {1}^{2} + {a}^{2} = \frac{ [ a \times ( a + 1 ) \times ( a + ( a + 1 ) ]}{6} . \\ \\ \sf \implies {1}^{2} + {2}^{2} + {a}^{2} = \frac{ [ a \times ( a + 1 ) \times ( a + ( a + 1 ) ]}{6}. \\ \\ \sf \implies {1}^{2} + {2}^{2} + {3}^{2} + {a}^{2} = \frac{ [ a \times ( a + 1 ) \times ( a + ( a + 1 ) ]}{6}.$

Now,

We have to find

(1)² + (2)² + (3)³ .... (9)² = $\sf \frac{9 \times 10 \times 19}{6} = 285$

Here a = 9 .

So,

Now,

→ (10)² + (11)² + (12)² + ..... (50)² .

Here a = 50 .

Then,

$\tiny \sf \rightarrow {(10)}^{2} + {(11)}^{2} + {(12)}^{2} + ...... + {(50)}^{2} . = \frac{50 \times (50 + 1) \times (50 + (50 + 1))}{6} - 285 . \\ \\ \sf = \frac{50 \times 51 \times 101}{6} - 285 \\ \\ \huge \pink{ \boxed{ \it = 42,640. }}$

Hence, it is solved .

Answer 2

Given

${1}^{2} + {2}^{2} = \frac{5 \times 3 \times 2}{6}$

${1}^{2} + {2}^{2} + {3}^{2} + {4}^{2} = \frac{4 \times 5 \times 9}{6} \\ \\ {1}^{2} + {2}^{2} + {3}^{2} + {4}^{2} = \frac{4 \times 5 \times 9}{6}$

Following This Series Things To Be Noted.

When we Divide 6 from Product in Fraction we get Sum of Squares of given Terms.

Like 1²=1+2²=4 1+4=5

2×3×5/6=30/6=5

We get Sum of Square.

Now we have to Find Value of

${10}^{2} + {11}^{2} + .... + {50}^{2}$

First Find The Sum of First 9 numbers Square.

1²=1,2²=4,3²=9,4²=16,5²=25,6²=36,7²=49,8²=64,9²=81.

There Sum

1+4+9+16+25+36+49+64+81=285

Now to Find Sum of Square of 1 to 50

Use Formula

$n(n + 1)(2n + 1) \div 6$

Where n=Total Number.

$50(50 + 1)(2 \times 50 + 1) \div 6 \\ \\ 50 \times 51 \times 101 \div 6 \\ \\ 42925$

Sum of First 50 numbers Square=42925

Sum of First 9 numbers Square=285

Sum of 10²+11²+...+50²=42925-285

=42640

For Finding Sum in Fraction

Let =Three Terms In Numerator=(a×b×c)

We know (a×b×c)÷6=Sum of The Numbers

$\frac{a \times b \times c}{6} = 42640 \\ \\ a \times b \times c = 42640 \times 6$

Prime Factors of

42640=2×2×2×2×5×13×41

$a \times b \times c = (2 \times 2 \times 2 \times 2 ) \times ( 5 \times 13) \times(41\times6) \\ \\ = 16 \times 65 \times246$

So

$\it{\large{ {10}^{2} + {11}^{2} + {12}^{2} + ... + {50}^{2} = 42640}}$

$\it{\large{{10}^{2} + {11}^{2} + {12}^{2} + ... + {50}^{2} = \frac{16 \times 65 \times 246}{6}} }$