Part 1Find the sum of the geometric series.343343negative 147−147plus+6363minus−midline ellipsis⋯ Part 1The sum of the series is [input]enter your response here . (Simplify your answer.)

Question

Part 1Find the sum of the geometric series.343343negative 147−147plus+6363minus−midline ellipsis⋯ Part 1The sum of the series is [input]enter your response here . ​(Simplify your​ answer.)

BlackTom AI · Accepted Answer

Since the terms 343, −147, 63, −27, … form a geometric sequence, the first term a = 343 and the common ratio r = (−147)/343 = −3/7.

Relevant concepts:
- An infinite geometric series with first term a and common ratio r converges only if |r| < 1.
- When it converges, the sum is S = a / (1 − r).

Step-by-step reasoning and calculations:
1. Identify a and r:
   a = 343
   r = −147 ÷ 343 = −3/7

2. Check convergence:
   |r| = 3/7 ≈ 0.4286, which is less than 1, so the infinite series converges.

3. Apply the sum formula:
   S = a / (1 − r)
   1 − r = 1 − (−3/7) = 1 + 3/7 = (7/7 + 3/7) = 10/7

So S = 343 ÷ (10/7) = 343 × (7/10)

4. Compute the product:
   343 × 7 = 2401
   2401 ÷ 10 = 240.1

Why this is correct:
- The ratio r is negative, so terms alternate signs; because |r| < 1 the alternating terms diminish in magnitude and the infinite sum converges to the finite value computed above.
- Using any incorrect r (for example a sign error giving r = +3/7 or a magnitude error) would produce a different denominator (1 − r) and thus the wrong sum. For instance, using r = +3/7 would give S = 343/(1 − 3/7) = 343/(4/7) = 343×7/4 = 600.25, which is not the series sum. Omitting the alternating sign or failing to check |r| < 1 would also be incorrect.

Conclusion:
The infinite geometric series sums to 240.1, which matches the provided answer.

Part 1Find the sum of the geometric series.343343negative 147−147plus+6363minus−midline ellipsis⋯ Part 1The sum of the series is [input]enter your response here . (Simplify your answer.)Multiple fill-in-the-blank

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