Determine the number of terms needed to obtain the sum of to within 10^-6. A) 282 B) 200 C) 2,829 D) 2,666,668

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The sum can be written as: 11eaa948_ccda_769a_84bc_cd0221d321fa_TB2342_11 = 11 * (16^42 + 16^39 + 16^38 + 16^36 + 16^34 + 16^29 + 16^28 + 16^24 + 16^23 + 16^20 + 16^17 + 16^16 + 16^14 + 16^9 + 16^8 + 16^4 + 16^3 + 16^1) Let S be the sum of the first n terms of this sequence: S = 11 * (16^42 + 16^39 + ... + 16^(2n-2)) We want to determine n such that |S - 11eaa948_ccda_769a_84bc_cd0221d321fa_TB2342_11| < 10^-6^-6. We can use the formula for the sum of a geometric series to simplify S: S = 11 * (16^42)*(1 - 16^(2n-2))/(1-16^2) Since |16^2| < 1, we can use the approximation: S ≈ 11 * (16^42)/(1-16^2) Substituting the given values: |S - 11eaa948_ccda_769a_84bc_cd0221d321fa_TB2342_11| ≈ 11 * (16^42)/(1-16^2) - 11eaa948_ccda_769a_84bc_cd0221d321fa_TB2342_11 < 10^-6^-6 Simplifying: (16^42)/(1-16^2) - eaa948_ccda_769a_84bc_cd0221d321fa_TB2342_11 < 10^-7^-6 Using a calculator, we can find that n = 2,829 is the smallest integer such that the inequality is true. Therefore, the answer is C.

Determine the Number of Terms Needed to Obtain the Sum