A large company put out an advertisement in a magazine for a job opening. The first day the magazine was published the company got 120 responses, but the responses were declining by 10% each day. Assuming the pattern continued, how many total responses would the company get over the course of the first 9 days after the magazine was published, to the nearest whole number? Answer:

Question

A large company put out an advertisement in a magazine for a job opening. The first day the magazine was published the company got 120 responses, but the responses were declining by  10% each day. Assuming the pattern continued, how many total responses would the company get over the course of the first 9 days after the magazine was published, to the nearest whole number? Answer:

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Identify initial responses and decrease: Identify the initial number of responses and the daily percentage decrease. Initial responses on the first day: 120 Daily decrease: 10%Calculate responses for each day: Calculate the number of responses for each day using the formula for the nth term of a geometric sequence, which is given by $ a_n = a_1 \times r^{(n-1)} $, where $ a_1 $ is the first term, $ r $ is the common ratio, and $ n $ is the term number. The common ratio $ r $ is 1 - 0.10 = 0.90 because the responses decrease by 10% each day.Calculate total responses over 9 days: Calculate the total number of responses over the 9 days by summing the geometric sequence. Total responses = $ a_1 + a_1 \times r + a_1 \times r^2 + \ldots + a_1 \times r^8 $Use formula for sum of geometric sequence: Use the formula for the sum of the first n terms of a geometric sequence, which is given by $ S_n = a_1 \times \frac{1 - r^n}{1 - r} $, where $ S_n $ is the sum of the first n terms. For the first 9 days, $ n = 9 $, so we calculate $ S_9 = 120 \times \frac{1 - 0.90^9}{1 - 0.90} $.Perform calculation for S_9: Perform the calculation for $ S_9 $. $ S_9 = 120 \times \frac{1 - 0.90^9}{1 - 0.90} $ $ S_9 = 120 \times \frac{1 - 0.387420489}{0.10} $ $ S_9 = 120 \times \frac{0.612579511}{0.10} $ $ S_9 = 120 \times 6.12579511 $ $ S_9 = 735.095412 $Round total responses: Round the total number of responses to the nearest whole number. Rounded total responses = 735 (to the nearest whole number)

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