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Question
Find (702 – 692) + (682 – 672) + (662 – 652) + ... + (22 – 12)
Solution
Let S = (702 – 692) + (682 – 672) + ... + (22 – 12)
∴ S = (22 – 12) + (42 – 32) + … + (702 – 692)
Here, 2, 4, 6, …, 70 is an A.P. with rth term = 2r
and 1, 3, 5, …, 69 in A.P. with rth term = 2r – 1
∴ S = `sum_("r" = 1)^35[(2"r")^2 - (2"r" - 1)^2]`
= `sum_("r" = 1)^35[4"r"^2 - (4"r"^2 - 4"r" + 1)]`
= `sum_("r" = 1)^35(4"r" - 1)`
= `4sum_("r" = 1)^35"r" - sum_("r" = 1)^35 1`
= `4.(35 xx 36)/2 - 35`
= (72 – 1) (35)
= 71 × 35
= 2485
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