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Find the following sums : (i) 12 + 13...

Find the following sums : (i) ` 12 + 13 + 14 + …+ 37` (ii) ` 18^(2) + 19^(2) + 20 ^(2) +…+ 47^(2)` (iii)` 2^(3) + 4^(3) + 6^(3) +… + 18^(3) `

Text Solution

Verified by Experts

(i) ` 12 + 13 + 14 +…+ 37 = (1 + 2 + 3 +…+ 37 ) - (1 + 2 + 3 + …+ 11) `
` = (37 (37 +1))/(2) - (11(11 +1))/(2)`
` = 703 - 66 = 637`
` (ii) 18^(2) + 19^(2) + 20^(2) +…+ 47^(2) = (1^(2) + 2^(2) + 3^(2) +…+ 47^(2)) - (1^(2) +2^(2) +3^(2) +...+ 17^(2))`
`(47xx(47 +1)xx(2xx47+1))/(6) - (17xx(17 +1)xx(2 xx 17 +1))/(6) `
` = (47xx 48 xx95)/(6) - (17xx 18xx 35)/(6)`
` 35720 - 1785 = 33935`
(iii) ` 2^(3) + 4^(3) + ...+ 18^(3) `
= `(2 xx 1)^(3) + (2 xx 2)^(3) + (2 xx 3)^(3) +...+ (2 xx9)^(3)`
` = 2^(3) (1^(3) + 2^(3) + 3^(3) +...+ 9^(3))`
` 8 xx[ (9 xx(9 +1))/(2)]^(2) = 16200`
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