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A large company put out an advertisement in a magazine for a job opening. The first day the magazine was published the company got 160 responses, but the responses were declining by 15% each day. Assuming the pattern continued, how many total responses would the company get over the course of the first 21 days after the magazine was published, to the nearest whole number? Answer:

A large company put out an advertisement in a magazine for a job opening. The first day the magazine was published the company got 160160 responses, but the responses were declining by 15% 15 \% each day. Assuming the pattern continued, how many total responses would the company get over the course of the first 2121 days after the magazine was published, to the nearest whole number?\newlineAnswer:

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Q. A large company put out an advertisement in a magazine for a job opening. The first day the magazine was published the company got 160160 responses, but the responses were declining by 15% 15 \% each day. Assuming the pattern continued, how many total responses would the company get over the course of the first 2121 days after the magazine was published, to the nearest whole number?\newlineAnswer:
  1. Identify Initial Data: Identify the initial number of responses and the daily decline rate.\newlineInitial responses on the first day: 160160\newlineDaily decline rate: 15%15\%\newlineWe need to calculate the total number of responses over 2121 days, considering a 15%15\% decline each day.
  2. Calculate Daily Responses: Calculate the number of responses for each day using the formula for a geometric sequence.\newlineThe number of responses on the nnth day can be calculated using the formula:\newlineResponsesn=Initial responses×(1−decline rate)(n−1)\text{Responses}_n = \text{Initial responses} \times (1 - \text{decline rate})^{(n-1)}\newlineWhere decline rate is 0.150.15 (15%15\% as a decimal).
  3. Calculate Total Responses: Calculate the total number of responses over 2121 days.\newlineWe will sum up the responses for each day using the formula from Step 22.\newlineTotal responses =Responsesday1+Responsesday2+…+Responsesday21= \text{Responses}_{\text{day}1} + \text{Responses}_{\text{day}2} + \ldots + \text{Responses}_{\text{day}21}\newlineWe will use a loop or a summation formula to calculate this total.
  4. Perform Daily Calculations: Perform the calculations for each day and sum them up.\newlineLet's start by calculating the first few terms to ensure our formula is correct.\newlineResponsesday1_{\text{day1}} = 160×(1−0.15)(1−1)=160×1=160160 \times (1 - 0.15)^{(1-1)} = 160 \times 1 = 160\newlineResponsesday2_{\text{day2}} = 160×(1−0.15)(2−1)=160×0.85=136160 \times (1 - 0.15)^{(2-1)} = 160 \times 0.85 = 136\newlineResponsesday3_{\text{day3}} = 160×(1−0.15)(3−1)=160×0.852≈115.6160 \times (1 - 0.15)^{(3-1)} = 160 \times 0.85^2 \approx 115.6\newline...\newlineWe will continue this process until we reach day 2121.
  5. Use Calculator for Sum: Use a calculator or a software tool to sum the geometric series.\newlineSince manually calculating each term and summing them up is time-consuming and prone to error, we will use a calculator or software to find the sum of the geometric series from day 11 to day 2121.
  6. Calculate Geometric Series Sum: Calculate the sum of the geometric series.\newlineThe sum of a geometric series can be calculated using the formula:\newlineSum = a×(1−rn)/(1−r)a \times (1 - r^n) / (1 - r)\newlineWhere:\newlinea = first term of the series (160160 responses)\newliner = common ratio (0.850.85, since the decline is 15\%))\(\newlinen = number of terms (\$21\) days)\(\newline\)Sum = \(160 \times (1 - 0.85^{21}) / (1 - 0.85)\)
  7. Evaluate Total Responses: Evaluate the sum to the nearest whole number.\(\newline\)Using the formula from Step \(6\), we calculate the sum.\(\newline\)Sum \(\approx 160 \times (1 - 0.85^{21}) / (1 - 0.85)\)\(\newline\)Sum \(\approx 160 \times (1 - 0.08589) / 0.15\)\(\newline\)Sum \(\approx 160 \times 0.91411 / 0.15\)\(\newline\)Sum \(\approx 975.05\)\(\newline\)Rounded to the nearest whole number, the total number of responses is approximately \(975\).

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