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An arithmetic progression has a first term of 160 and a common difference of -8. What is the general term and the 8th term?

Mathematics

ScanSolve AI Answer

Answer

an=(168−8n) mmHga_n = (168 - 8n) \text{ mmHg}

104 mmHg104 \text{ mmHg}

Step-by-step solution

5 steps
  1. Analyze the given sequence and identify the pattern.

    The sequence of systolic BP readings is 160,152,148,136,128 mmHg160, 152, 148, 136, 128 \text{ mmHg}.
    Let's find the differences between consecutive terms:
    152−160=−8152 - 160 = -8
    148−152=−4148 - 152 = -4
    136−148=−12136 - 148 = -12
    128−136=−8128 - 136 = -8
    The differences are not constant, so the sequence as written is not a standard arithmetic progression. However, given the context of arithmetic progressions, it is highly probable that there is a typo in the sequence. The common difference appears to be −8 mmHg-8 \text{ mmHg} for most terms. We will assume the sequence is an arithmetic progression with a common difference of −8 mmHg-8 \text{ mmHg}, meaning the third term 148 would ideally be 144 to fit this pattern.

  2. Identify the first term and common difference.

    Based on our assumption, the first term is a1=160 mmHga_1 = 160 \text{ mmHg}.
    The common difference is d=−8 mmHgd = -8 \text{ mmHg}.

  3. Write the general term for the arithmetic progression.

    The formula for the nn-th term of an arithmetic progression is an=a1+(n−1)da_n = a_1 + (n-1)d.
    Substitute a1=160a_1 = 160 and d=−8d = -8:

    an=160+(n−1)(−8)a_n = 160 + (n-1)(-8)
  4. Simplify the general term.

    an=160−8n+8a_n = 160 - 8n + 8 an=168−8na_n = 168 - 8n

    The general term is an=(168−8n) mmHg\boxed{a_n = (168 - 8n) \text{ mmHg}}.

  5. Predict the BP after 8 hours.

    The question asks for the BP after 8 hours, which corresponds to the 8th term (n=8n=8) in the sequence.
    Substitute n=8n=8 into the general term formula:

    a8=168−8(8)a_8 = 168 - 8(8) a8=168−64a_8 = 168 - 64 a8=104 mmHga_8 = 104 \text{ mmHg}

    The predicted BP after 8 hours is 104 mmHg\boxed{104 \text{ mmHg}}.

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