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Question Number 186631 by DAVONG last updated on 07/Feb/23

Commented by mr W last updated on 08/Feb/23

$a l l a n s w e r s g i v e n a r e w r o n g .$ $a_{k} \in N \Rightarrow a_{k} ⩾ 1$ $s i n c e$ $1^{2} + 2^{2} + 3^{2} + . . . + 50^{2} = \frac{50 \times 51 \times 101}{6} = 42925$ ${(a_{1})}^{2} + {(2 a_{2})}^{2} + {(3 a_{3})}^{2} + . . . {(50 a_{50})}^{2} = 42925$ $h a s o n l y o n e s o l u t i o n :$ $a_{1} = a_{2} = a_{3} = . . . = a_{50} = 1$ $s o a_{1} + a_{2} + a_{3} + . . . + a_{50} i s a n d c a n o n l y$ $b e e q u a l t o 50 .$
Commented by mr W last updated on 09/Feb/23

$c o r r e c t q u e s t i o n s e e Q 186732$
	Answered by mahdipoor last updated on 07/Feb/23
	$B={a_1 ,a_2 ,...,a_(50) } , a_i ∈N f(B)=(a_1 )^2 +(2a_2 )^2 +...+(50a_(50) )^2 for B_1 ={1,1,...,1} , f(B_1 )=1^2 +2^2 +...+50^2 = (((50)×(50+1)×(2×50+1))/6)=42925 ⇒a_1 +a_2 +...+a_(50) =1+2+...+50=((50×51)/2)= 1275$
	$B = {a_{1}, a_{2}, . . ., a_{50}}, a_{i} \in N$ $f (B) = {(a_{1})}^{2} + {(2 a_{2})}^{2} + . . . + {(50 a_{50})}^{2}$ $f o r B_{1} = {1, 1, . . ., 1}, f (B_{1}) = 1^{2} + 2^{2} + . . . + 50^{2} =$ $\frac{(50) \times (50 + 1) \times (2 \times 50 + 1)}{6} = 42925$ $\Rightarrow a_{1} + a_{2} + . . . + a_{50} = 1 + 2 + . . . + 50 = \frac{50 \times 51}{2} =$ $1275$
	Commented by mr W last updated on 07/Feb/23

	$t h e q u e s t i o n i s s t r a n g e .$ $w h e n a_{1} = a_{2} = a_{3} = . . . = a_{50} = 1 s u c h t h a t$ ${(1 \times 1)}^{2} + {(2 \times 1)}^{2} + {(3 \times 1)}^{2} + . . . + {(50 \times 1)}^{2} = 42925,$ $t h e n a_{1} + a_{2} + a_{3} + . . . + a_{50} = 50, n o t$ $1275 .$ $w h e n a_{1} = 1, a_{2} = 2, . . ., a_{50} = 50 s u c h t h a t$ $a_{1} + a_{2} + a_{3} + . . . + a_{50} = \frac{50 \times 51}{2} = 1275,$ $t h e n {(a_{1})}^{2} + {(2 a_{2})}^{2} + . . . + {(50 a_{50})}^{2} =$ $1^{4} + 2^{4} + 3^{4} + . . . + 50^{4}$ $= \frac{50 \times 51 \times 101 \times 7649}{30} = 65666765$ $\neq 42925$
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