Applications of AP MCQs Quiz | Class 10

Understanding Applications of Arithmetic Progressions (AP)

Arithmetic Progressions (AP) are sequences of numbers where the difference between consecutive terms is constant. This constant difference is known as the common difference. APs are not just abstract mathematical concepts; they have numerous practical applications in daily life and various fields.

Key Concepts in AP

Before diving into applications, let’s quickly review the fundamental formulas:

• First term (a): The starting term of the sequence.
• Common difference (d): The constant difference between any two consecutive terms.
• Number of terms (n): The total count of terms in the sequence.
• nth term (an): The value of the term at a specific position ‘n’.
  Formula: an = a + (n - 1)d
• Sum of n terms (Sn): The total sum of all terms up to the nth term.
  Formula: Sn = n/2 [2a + (n - 1)d] or Sn = n/2 [a + an]

Real-Life Applications of AP Word Problems

Word problems involving AP often describe situations where a quantity increases or decreases by a fixed amount at regular intervals. Here are common scenarios:

1. Salary Increments: If a person’s salary increases by a fixed amount each year, their annual salaries form an AP. We can use the nth term formula to find the salary after a certain number of years or the sum formula to find their total earnings over a period.
2. Loan Repayments: When a loan is repaid in installments where the principal amount decreases by a constant value each month (or year), the outstanding amounts or the interest paid can form an AP. The sum formula can help calculate total interest or total repayment.
3. Production/Sales Growth: A company’s production or sales figures might increase by a steady number of units each year. We can use AP formulas to predict future production or calculate total production over several years.
4. Construction and Design:
   • Ladder Rungs: The length of rungs of a ladder often decreases uniformly from bottom to top.
   • Stadium Seats: The number of seats in rows of a stadium typically increases by a constant number for each successive row.
5. Savings Schemes: If someone saves a fixed amount more each month than the previous month, their monthly savings form an AP. The sum formula helps calculate total savings over a period.
6. Distance Covered: Problems where an object covers increasing or decreasing distances by a fixed amount in successive time intervals.

How to Approach AP Word Problems

Solving word problems involving AP requires careful reading and identification of the given values:

1. Identify ‘a’: Determine the first term (initial value).
2. Identify ‘d’: Find the common difference (the fixed increase or decrease).
3. Identify ‘n’: Determine the number of terms or the specific position you are interested in.
4. Identify what to find: Is the problem asking for the nth term (an) or the sum of n terms (Sn)?
5. Apply the correct formula: Substitute the identified values into the appropriate AP formula and solve.

Quick Revision Checklist

• First term = a
• Common difference = d
• Number of terms = n
• nth term = a + (n-1)d
• Sum of n terms = n/2 [2a + (n-1)d] or n/2 [a + an]

Practice Questions

Test your understanding with these additional practice problems:

1. A sum of Rs. 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs. 20 less than its preceding prize, find the value of each of the prizes.
2. In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato, and other potatoes are placed 3 m apart in a straight line. There are 10 potatoes in the line. A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the second potato, and so on. What is the total distance the competitor has to run?
3. A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: Rs. 200 for the first day, Rs. 250 for the second day, Rs. 300 for the third day, etc., the penalty for each succeeding day being Rs. 50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days?
4. The production of TV sets in a factory increases uniformly by a fixed number every year. It produced 16000 sets in the 6th year and 22000 sets in the 9th year. Find the production in the 1st year and in the 10th year.
5. How many terms of the AP: 9, 17, 25, … must be taken to give a sum of 636?