Question

Average of 45 consecutive number is 398 and 45 consecutive odd number is 499 .what is square diff b/w largest even number and smallest odd number

Answer 1

Answers

$square \: of \: difference \: between \: largest \: even \: number \: and \: smallest \: odd \: number \: is \: 1225.$

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Step-by-step explanation:

$Let \: 45 \: consecutive \: numbers \: be \: x, \: (x + 1), \: (x + 2), \: (x + 3), \: (x + 4)..............(x + 44). \\ Average \: of \: 45 \: consecutive \: number \: is \: 398 \\ </p><p>Therefore \\ \frac{x + (x + 1) + (x + 2) + (x + 3) + ......... + (x + 43) + (x + 44)}{45} = 398 \\ \frac{45x + (1 + 2 + 3 + ...... + 44)}{45} = 398 \\ \\ As \: summation \: of \: n \: consecutive \: integers \: is \: given \: as \\ S = \frac{n(n + 1)}{2} \\ Therefore \: summation \: of \: 44 \: consecutive \: integers \: is \\ S = \frac{44(44 + 1)}{45} \\ (22)(45) \\ Therefore \\ \frac{45x + (1 + 2 + 3 + ...... + 44)}{45} = \frac{45x + (22)(45)}{45} = 398 \\ \frac{45(x + 22}{45} = 398 \\ x + 22 = 398 \\ x = 376 \\ x + 44 = 376 + 44 \\ (x + 44) = 420 \\ Hence \: largest \: integer \: of \: 45 \: consecutive \: number \: is \: (x + 44) = 420. \\ \\ Let \: 45 \: consecutive \: odd \: numbers \: be \: x, \: (x + 2), \: (x + 4), \: (x + 6), \: (x + 8)..............(x + 88). \\ Average \: of \: 45 \: consecutive \: odd \: number \: is \: 499 \\ Therefore \\ \frac{x + (x + 2) + (x + 4) + (x + 6) + ...... + (x + 88) }{45} = 499 \\ \frac{45x + (2 + 4 + 6 + ........ + 88)}{45} = 499 \\ Now \: ( 2, \: 4, \: 6, \: 8,,,,,,,,,,,88) \: is \: an \: AP \: (2 + 4 + 6 + ........ + 88) \: with \: a = 2 \: and \: d = 2 </p><p>Therefore, \: (2 + 4 + 6 + ........ + 88) \: is \: summation \: of \: AP \: with \: a = 2 \: and \: d = 2 \: and \: n = 44 \\ Therefore \: by \: summation \: formula \: for \: AP: Sn = \frac{n}{2} ((2a) + (n - 1)d) \\ Hence \: S(44) = \frac{44}{2} (2(2) + (44 - 1)(2)) \\ S(44) = (22)(4 + (43)(2)) \\ S(44) = (22)(90) \\ Hence \\ \frac{45x + (2 + 4 + 6 + ........ + 88)}{45} = \frac{45x + (22)(90)}{45} = 499 \\ \frac{45(x + (22)(2)}{45} = 499 \\ (x + 44) = 499 \\ (x + 44) - 44 = 499 - 44 \\ x = 455 \\ Now \: x \: is \: smallest \: term \: in \: 45 \: consecutive \: odd \: numbers \\ </p><p>Therefore \: smallest \: number \: in \: consecutive \: odd \: numbers \: is \: 455. \\ \\ \\ Now \: largest \: even \: number \: is \: 420 \: and \: smallest \: odd \: number \: is \: 455 \\ </p><p>Therefore, \: square \: of \: difference \: between \: largest \: even \: number \: and \: smallest \: odd \: number \: is: \\ {(455 - 420)}^{2} = {35}^{2} = 1225. \\ Hence \: square \: of \: difference \: between \: largest \: even \: number \: and \: smallest \: odd \: number \: is \: 1225.$

Answer 2

Answer:

158

Step-by-step explanation: