NCERT Solutions & Concepts: Arithmetic Progressions Ex 5.3

Welcome! In the previous exercises, you mastered finding any specific term in an Arithmetic Progression (AP) using the a_n formula. Now, we'll level up and tackle a new challenge: finding the sum of a certain number of terms in an AP. This is the core focus of Exercise 5.3. Imagine calculating the total money saved over a year if you save ₹100 in the first month, ₹120 in the second, ₹140 in the third, and so on. Instead of adding each month's savings manually, you can use a powerful formula. In this chapter guide, you will learn the two key formulas for the sum of the first 'n' terms of an AP, understand how to apply them in different scenarios, and solve a variety of problems, including word problems, just like the ones you'll face in your CBSE board exams.

Understanding the Sum of an AP

Finding the sum of an AP might seem like a lot of addition, but there's a clever trick. Let's say we want to sum the numbers from 1 to 10. The AP is 1, 2, 3, ..., 10.
Let's write the sum, S, twice:
S = 1 + 2 + 3 + ... + 9 + 10
S = 10 + 9 + 8 + ... + 2 + 1

Now, add these two equations vertically:
2S = (1+10) + (2+9) + (3+8) + ... + (9+2) + (10+1)
2S = 11 + 11 + 11 + ... + 11 + 11
Since there are 10 terms, we have:
2S = 10 × 11
S = (10 × 11) / 2 = 55

This simple idea is generalized for any AP. The sum of the first n terms, denoted as S_n, is given by the formula: S_n = n/2 [2a + (n-1)d]. Here, 'a' is the first term, 'd' is the common difference, and 'n' is the number of terms you want to add. This formula saves you from the tedious task of manual addition, especially when 'n' is large.

Key Formulas for Sum of an AP (S_n)

General Formula (when a, n, and d are known)
The primary formula to find the sum is: S_n = n/2 [2a + (n-1)d]. Where: S_n is the sum of the first n terms. n is the number of terms. a is the first term. d is the common difference.
Formula with Last Term (when a, n, and l are known)
If you know the first term (a), the last term (l or a_n), and the number of terms (n), you can use this simpler formula: S_n = n/2 [a + l]. This is derived from the main formula, as 2a + (n-1)d can be written as a + [a + (n-1)d], and we know l = a_n = a + (n-1)d.

Step-by-Step Solved Examples

  • Example 1: Find the sum of the first 22 terms of the AP: 8, 3, -2, ... Step 1: Identify the given values. From the AP, the first term a = 8. The common difference d = 3 - 8 = -5. The number of terms to sum is n = 22. Step 2: Choose the correct formula. We have a, d, and n. So, we will use the formula S_n = n/2 [2a + (n-1)d]. Step 3: Substitute the values and calculate. S_22 = 22/2 [2(8) + (22-1)(-5)] S_22 = 11 [16 + (21)(-5)] S_22 = 11 [16 - 105] S_22 = 11 [-89] S_22 = -979 Final Answer: The sum of the first 22 terms of the given AP is -979.
  • Example 2: Find the sum of the AP: 7 + 10.5 + 14 + ... + 84. Step 1: Identify the given values. Here, the first term a = 7. The last term l = a_n = 84. The common difference d = 10.5 - 7 = 3.5. The number of terms n is unknown. We cannot find the sum without n. Step 2: Find the number of terms (n). We use the formula for the nth term: a_n = a + (n-1)d. 84 = 7 + (n-1)(3.5) 84 - 7 = (n-1)(3.5) 77 = (n-1)(3.5) n-1 = 77 / 3.5 n-1 = 22 n = 23 Step 3: Find the sum (S_n). Now that we know n = 23, a = 7, and l = 84, we can use the simpler sum formula: S_n = n/2 [a + l]. S_23 = 23/2 [7 + 84] S_23 = 23/2 [91] S_23 = 2093 / 2 = 1046.5 Final Answer: The sum of the given AP is 1046.5.

Exam Tips & Common Mistakes

Be careful during your exams! Here are some common pitfalls to avoid:

  • Confusing a_n and S_n: Always read the question carefully. Are you asked to find a specific term (a_n) or the sum of terms (S_n)? Students often mix up their formulas. a_n = a + (n-1)d is for one term, while S_n = n/2 [2a + (n-1)d] is for the sum.
  • Sign Errors with 'd': When the AP is decreasing, the common difference d is negative. It's very easy to make a calculation mistake with these negative signs. Double-check your multiplication.
  • Forgetting to find 'n': In problems like 3 + 7 + 11 + ... + 99, the last term is given, not n. You must first use the a_n formula to find the value of n before you can calculate the sum S_n.
  • Incorrectly Identifying 'a': In some word problems, the 'first term' might correspond to the first year, first day, or first installment. Make sure you correctly identify what a and n represent in the context of the problem.

Practice Questions with Solutions

  • Q: Find the sum of the first 40 positive integers divisible by 6. A: Step 1: Identify the AP. The first positive integer divisible by 6 is 6. The second is 12, and so on. The AP is 6, 12, 18, ... Here, a = 6, d = 6, and we need to find the sum of the first n = 40 terms. Step 2: Use the sum formula S_n = n/2 [2a + (n-1)d]. S_40 = 40/2 [2(6) + (40-1)6] S_40 = 20 [12 + (39)6] S_40 = 20 [12 + 234] S_40 = 20 [246] S_40 = 4920 Final answer: The sum of the first 40 positive integers divisible by 6 is 4920.
  • Q: In an AP, a = 5, d = 3, and a_n = 50. Find n and S_n. A: Step 1: Find n using the a_n formula. We are given a = 5, d = 3, a_n = 50. a_n = a + (n-1)d 50 = 5 + (n-1)3 45 = (n-1)3 n-1 = 45/3 = 15 n = 16 Step 2: Find S_n using the formula S_n = n/2 [a + l], where l = a_n = 50. S_16 = 16/2 [5 + 50] S_16 = 8 [55] S_16 = 440 Final answer: n = 16 and S_n = 440.
  • Q: Find the sum of the first 15 multiples of 8. A: Step 1: Form the AP. The multiples of 8 are 8, 16, 24, ... This is an AP with first term a = 8, common difference d = 8, and the number of terms is n = 15. Step 2: Apply the sum formula S_n = n/2 [2a + (n-1)d]. S_15 = 15/2 [2(8) + (15-1)8] S_15 = 15/2 [16 + (14)8] S_15 = 15/2 [16 + 112] S_15 = 15/2 [128] S_15 = 15 * 64 S_15 = 960 Final answer: The sum of the first 15 multiples of 8 is 960.
  • Q: The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference. A: Step 1: Use the sum formula S_n = n/2 [a + l] to find n. We are given a = 5, l = 45, and S_n = 400. 400 = n/2 [5 + 45] 400 = n/2 [50] 400 = 25n n = 400 / 25 = 16. Step 2: Use the nth term formula l = a + (n-1)d to find d. We now know n=16. 45 = 5 + (16-1)d 40 = 15d d = 40 / 15 = 8 / 3 Final answer: The number of terms n is 16 and the common difference d is 8/3.

Frequently Asked Questions

What is the difference between a_n and S_n in an AP?

`a_n` represents the value of a single, specific term at the nth position in the sequence. `S_n` represents the total sum of all terms from the first term up to and including the nth term.

When should I use S_n = n/2 [2a + (n-1)d] vs S_n = n/2 [a + l]?

Use `S_n = n/2 [2a + (n-1)d]` when you know the first term (a), common difference (d), and number of terms (n). Use the shortcut `S_n = n/2 [a + l]` when you know the first term (a), the last term (l), and the number of terms (n), as it is faster.

Can the sum of an Arithmetic Progression be negative?

Yes, absolutely. If an AP has negative terms, or if it's a decreasing sequence that goes from positive to negative, the sum of its terms can be negative or even zero.