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Find the sum of the first 7 terms in the following geometric series. Do not round your answer. 1024+512+256+dots

Find the sum of the first 77 terms in the following geometric series. Do not round your answer.\newline1024+512+256+… 1024+512+256+\ldots

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Full solution

Q. Find the sum of the first 77 terms in the following geometric series. Do not round your answer.\newline1024+512+256+… 1024+512+256+\ldots
  1. Identify first term and common ratio: To find the sum of the first 77 terms of a geometric series, we need to identify the first term (a1a_1) and the common ratio (rr) of the series.\newlineThe first term a1a_1 is 10241024.\newlineTo find the common ratio, we divide the second term by the first term: r=5121024=12r = \frac{512}{1024} = \frac{1}{2}.
  2. Calculate the common ratio: Now that we have the first term and the common ratio, we can use the formula for the sum of the first nn terms of a geometric series: Sn=a1×(1−rn)/(1−r)S_n = a_1 \times (1 - r^n) / (1 - r), where nn is the number of terms.\newlineWe want to find the sum of the first 77 terms, so n=7n = 7.
  3. Use the formula for sum of first n terms: Let's plug the values into the formula: S7=1024×(1−(1/2)7)/(1−1/2)S_7 = 1024 \times (1 - (1/2)^7) / (1 - 1/2).\newlineNow we calculate (1/2)7(1/2)^7.
  4. Plug in values into the formula: (12)7=1128(\frac{1}{2})^7 = \frac{1}{128}.\newlineNow we substitute this value back into the sum formula: S7=1024×(1−1128)/(1−12)S_7 = 1024 \times (1 - \frac{1}{128}) / (1 - \frac{1}{2}).
  5. Calculate (1/2)7(1/2)^7: Simplify the expression inside the parentheses: 1−1/128=127/1281 - 1/128 = 127/128. Now the sum formula looks like this: S7=1024×(127/128)/(1/2)S_7 = 1024 \times (127/128) / (1/2).
  6. Substitute value into sum formula: We simplify the denominator: 1−12=121 - \frac{1}{2} = \frac{1}{2}. Now we have S7=1024×(127128)/(12)S_7 = 1024 \times \left(\frac{127}{128}\right) / \left(\frac{1}{2}\right).
  7. Simplify expression inside parentheses: To divide by 12\frac{1}{2}, we multiply by the reciprocal, which is 22. So, S7=1024×(127128)×2S_7 = 1024 \times \left(\frac{127}{128}\right) \times 2.
  8. Simplify the denominator: Now we perform the multiplication: 1024×2=20481024 \times 2 = 2048. Then we multiply this by 127/128127/128: S7=2048×(127/128)S_7 = 2048 \times (127/128).
  9. Multiply by reciprocal: Finally, we calculate the multiplication: 2048×(127/128)=2048×127/1282048 \times (127/128) = 2048 \times 127 / 128.
  10. Perform multiplication: Perform the division: 2048×127/128=2048/128×1272048 \times 127 / 128 = 2048 / 128 \times 127.\newline2048/128=162048 / 128 = 16.\newlineSo, S7=16×127S_7 = 16 \times 127.
  11. Perform division: Now we multiply 1616 by 127127 to get the final sum: S7=16×127=2032S_7 = 16 \times 127 = 2032.

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