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Let = 3+4+8+9+13+14+ upto 40 terms. If ( )^ 1020 is a root of the equation x^2+x-2=0 , (0, 2 ) , then ^2 + 3 ^2 is equal to:

Options

  1. A2
  2. B7 4
  3. C5 2
  4. D3 2

Correct answer

A. 2

Step-by-step solution

The given series is = 3+4+8+9+13+14+ up to 40 terms. We can split the series into two arithmetic progressions, each containing 20 terms: S₁ = 3 + 8 + 13 + up to 20 terms S₂ = 4 + 9 + 14 + up to 20 terms Using the sum formula for an AP, S_n = n 2 [2a + (n-1)d] : S₁ = 20 2 [2(3) + (20-1)5] = 10[6 + 95] = 1010 S₂ = 20 2 [2(4) + (20-1)5] = 10[8 + 95] = 1030 Therefore, = S₁ + S₂ = 1010 + 1030 = 2040 . The exponent in the given expression is 1020 = 2040 1020 = 2 . Thus, ( )^2 = ^2 is a root of the equation x^2+x-2=0 . So

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