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प्रश्न
Answer the following:
Find (502 – 492) + (482 – 472) + (462 – 452) + ... + (22 – 12)
उत्तर
(502 – 492) + (482 – 472) + (462 – 452) + ... + (22 – 12)
= (502 + 482 + 462 + … + 22) – (492 + 472 + 452 + …+ 12)
= `sum_("r" = 1)^25 (2"r")^2 - sum_("r" = 1)^25(2"r" - 1)^2`
= `sum_("r" = 1)^25 4"r"^2 - sum_("r" = 1)^25(4"r"^2 - 4"r" + 1)`
= `sum_("r" = 1)^25[4"r"^2 - (4"r"^2 - 4"r" + 1)]`
= `sum_("r" = 1)^25(4"r" - 1)`
= `4sum_("r" = 1)^25 "r" - sum_("r" = 1)^25 1`
= `4 xx (25(25 + 1))/2 - 25`
= `(4(25)(26))/2 - 25`
= 1300 – 25
= 1275
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