Q. 1+2+3+4+....+20 = ? // Q. 1^2+2^2+3^2+.......+20^2 = ? // Q. 1^3+2^3+3^3......+20^3 = ? :: Arithmetic Series : Fast Calculation / āĻĻā§ā§°ā§āĻ¤ āĻāĻŖāĻ¨āĻž
Q. 1+2+3+4+..............+20 = ? āĻĒā§ā§°āĻļā§āĻ¨: ā§§+ā§¨+ā§Š+ā§Ē+â¯+ā§¨ā§Ļ = āĻāĻŋāĻŽāĻžāĻ¨ ?
Formula / āĻ¸ā§āĻ¤ā§ā§°: Sum = n(n+1) / 2
Solution / āĻ¸āĻŽāĻžāĻ§āĻžāĻ¨:
= 20 (20+1) / 2
= 20*21 /2 = 10*21 = 210
Trick / āĻā§āĻļāĻ˛: ā§¨ā§Ļ āĻ ā§¨ ā§°ā§ āĻāĻžāĻ āĻā§°āĻŋāĻ˛ā§ ā§§ā§Ļ āĻĒā§ā§ąāĻž āĻ¯āĻžāĻ¯āĻŧāĨ¤ āĻ¤āĻžā§° āĻĒāĻŋāĻāĻ¤ ā§§ā§Ļ × ā§¨ā§§ = ā§¨ā§§ā§ĻāĨ¤ (Divide 20 by 2 to get 10, then multiply by 21. The answer is 210.)
Q. 12+22+32+42+..............+202 = ? āĻĒā§ā§°āĻļā§āĻ¨: ā§§²+ā§¨²+ā§Š²+ā§Ē²+â¯+ā§¨ā§Ļ² = āĻāĻŋāĻŽāĻžāĻ¨ ?
Formula / āĻ¸ā§āĻ¤ā§ā§°: 12+22+32+42+...............+ n2 = n(n+1)(2n+1)/6
Solution / āĻ¸āĻŽāĻžāĻ§āĻžāĻ¨:
āĻāĻ¯āĻŧāĻžāĻ¤ n = 20
= 20 * 21 * 41/ 6
= 10 * 21 * 41/ 3
= 10 * 7 * 41
= 2870
Ans / āĻāĻ¤ā§āĻ¤ā§°: ā§¨ā§Žā§ā§Ļ
Trick / āĻā§āĻļāĻ˛: 20 * 21 * 41/ 6 = 2870
Note: To find the sum of the squares from 1 to 20, use the formula : n(n+1)(2n+1)/6
(ā§§ ā§° āĻĒā§°āĻž ā§¨ā§Ļ āĻ˛ā§āĻā§ āĻĒā§ā§°āĻ¤āĻŋāĻā§ āĻ¸āĻāĻā§āĻ¯āĻžā§° āĻŦā§°ā§āĻā§° āĻ¯ā§āĻāĻĢāĻ˛ āĻāĻ˛āĻŋāĻ¯āĻŧāĻžāĻŦāĻ˛ā§ n(n+1)(2n+1)/6 āĻ¸ā§āĻ¤ā§ā§°āĻā§ āĻŦā§āĻ¯ā§ąāĻšāĻžā§° āĻā§°āĻž āĻšāĻ¯āĻŧāĨ¤)
Q. 13+23+33+43+..............+203 = ? āĻĒā§ā§°āĻļā§āĻ¨: ā§§ā§Š +ā§¨ā§Š+ā§Šā§Š+ā§Ēā§Š+â¯+ā§¨ā§Ļā§Š = āĻāĻŋāĻŽāĻžāĻ¨ ?
Formula / āĻ¸ā§āĻ¤ā§ā§°: 13+23+33+43+..............+n3 = [n(n+1) /2]2
Solution / āĻ¸āĻŽāĻžāĻ§āĻžāĻ¨:
āĻāĻ¯āĻŧāĻžāĻ¤ n = 20
= {20*21/2 }2
= {10*21}2
= 2102
= 44100
Ans / āĻāĻ¤ā§āĻ¤ā§°: ā§Ēā§Ē,ā§§ā§Ļā§Ļ
Trick / āĻā§āĻļāĻ˛: āĻĒā§ā§°āĻĨāĻŽā§: 20*21/2 = 210, āĻ¤āĻžā§° āĻĒāĻŋāĻāĻ¤: 210*210 = 44100
The sum of cubes from 1 to 20 equals the square of the sum of numbers from 1 to 20. (ā§§ ā§° āĻĒā§°āĻž ā§¨ā§Ļ āĻ˛ā§āĻā§ āĻ¸āĻāĻā§āĻ¯āĻžāĻŦā§ā§°ā§° āĻāĻ¨ā§° āĻ¯ā§āĻāĻĢāĻ˛ = āĻ¸ā§āĻ āĻ¸āĻāĻā§āĻ¯āĻžāĻŦā§ā§°ā§° āĻ¯ā§āĻāĻĢāĻ˛ā§° āĻŦā§°ā§āĻāĨ¤)
āĻ ā§°ā§āĻĨāĻžā§, 13+23+33+43+..............+203 = (1 + 2 +.............+ 20)2 = 2102 = 44100