Exams Joint Entrance Examination Mathematics Statistics Question 119 Question 119 of 208 Medium NUMERICAL
If the variance of the first n natural numbers is 10 and the variance of the first m even natural numbers is 16, then m + n m+n m + n is equal to \_\_\_\_\_\_\_\_.
Answer: 18
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Var ( 1 , 2 , … , n ) = n 2 − 1 12 = 10 ⟹ n 2 − 1 = 120 ⟹ n 2 = 121 ⟹ n = 11 \text{Var}(1,2,\ldots,n) = \dfrac{n^2-1}{12}=10 \implies n^2-1=120 \implies n^2=121 \implies n=11 Var ( 1 , 2 , … , n ) = 12 n 2 − 1 = 10 ⟹ n 2 − 1 = 120 ⟹ n 2 = 121 ⟹ n = 11
Var ( 2 , 4 , … , 2 m ) = 4 ⋅ Var ( 1 , … , m ) = 4 ⋅ m 2 − 1 12 = 16 ⟹ m 2 − 1 = 48 ⟹ m 2 = 49 ⟹ m = 7 \text{Var}(2,4,\ldots,2m) = 4\cdot\text{Var}(1,\ldots,m) = 4\cdot\dfrac{m^2-1}{12}=16 \implies m^2-1=48 \implies m^2=49 \implies m=7 Var ( 2 , 4 , … , 2 m ) = 4 ⋅ Var ( 1 , … , m ) = 4 ⋅ 12 m 2 − 1 = 16 ⟹ m 2 − 1 = 48 ⟹ m 2 = 49 ⟹ m = 7
m + n = 7 + 11 = 18 m+n = 7+11=18 m + n = 7 + 11 = 18 Answer the question to unlock