express 441 , as the sum of 21 odd numbers
Answers
Answer:
As 441 = 21² it can be expressed as
1+3+5+7+9+11+13+15+17+19+21+23+25+27+29+31+33+35+37+39+41
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Answer:
14² + 15² = 421
15² + 16² = 481
Step-by-step explanation:
Express 441 square number as the sum of two consecutive natural numbers
Let say two consecutive natural numbers are
n & n+1
n² + (n+1)² = 441
=> n² + n² + 1 + 2n = 441
=> 2n² + 2n - 440 = 0
dividing by 2 both sides
=> n² + n - 220 = 0
There are no proper factor
So question can be either 421 or 481
lets take first 421
then
n² + n² + 1 + 2n = 421
=> 2n² + 2n - 420 = 0
dividing by 2 both sides
=> n² + n - 210 = 0
=> n² + 15n - 14n - 210 = 0
=> n(n+15)-14(n+15) = 0
=> (n-14)(n+15) = 0
n = 14 n = -15 (-ve integer not possible)
n+1 = 15
14² + 15² = 421
Similarly taking 481
n² + n² + 1 + 2n = 481
=> 2n² + 2n - 480 = 0
dividing by 2 both sides
=> n² + n - 240 = 0
=> n² + 16n - 15n - 210 = 0
=> n(n+16)-15(n+16) = 0
=> (n-15)(n+16) = 0
n = 15 n = -16 (-ve integer not possible)
n+1 = 16
15² + 16² = 481
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