Math miscellaneousHard
Question
13 - 23 + 33 - 43 +.....+93 =
Options
A.425
B.- 425
C.475
D.- 475
Solution
13 - 23 + 33 - 43 +.........+ 93
= 13 + 33 + 53 + ..... + 93 - (23 + 43 +......+83)
= S1 - S2
For S1, tn = (2n - 1)3 = 8n3 - 12n3 + 6n - 1
S1 = Σtn = 8Σn3 - 12Σn2 + 6Σn - Σ1
Here n = 5. Hence S1 = 2 × 25 × 36 - 2 × 5 × 6 × 11 + 3 × 30 - 5
= 1800 - 660 + 90 - 5 = 1890 - 665 = 1225
For S2, tn = 8n3 ; S2 = Σtn = 8Σn3 =
= 2 × 16 × 25 = 800. (for n = 4)
∴ Required sum = 1225 - 800 = 425.
= 13 + 33 + 53 + ..... + 93 - (23 + 43 +......+83)
= S1 - S2
For S1, tn = (2n - 1)3 = 8n3 - 12n3 + 6n - 1
S1 = Σtn = 8Σn3 - 12Σn2 + 6Σn - Σ1
Here n = 5. Hence S1 = 2 × 25 × 36 - 2 × 5 × 6 × 11 + 3 × 30 - 5
= 1800 - 660 + 90 - 5 = 1890 - 665 = 1225
For S2, tn = 8n3 ; S2 = Σtn = 8Σn3 =
= 2 × 16 × 25 = 800. (for n = 4) ∴ Required sum = 1225 - 800 = 425.
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