CBSE Class 10th Sum of First n Terms of an AP Details & Preparations Downloads
Arithmetic Progression (AP) is a fundamental concept in mathematics that finds applications in various fields, from finance to physics. One of the key aspects of an AP is its ability to generate a sequence of numbers with a constant difference between consecutive terms. In this blog post, we will delve into the intriguing world of APs and specifically focus on unraveling the formula for the Nth term.
Harmony in Numbers Demystifying the Sum of First n Terms in Arithmetic Progressions
Sum of N Terms Formula
The sum of n terms of AP is the sum(addition) of the first n terms of the arithmetic sequence. It is equal to n divided by 2 times the sum of twice the first term – ‘a’ and the product of the difference between second and first termed’ also known as common difference, and (n-1), where n is a number of terms to be added.
Sum of n terms of AP = n/2[2a + (n – 1)d]
For example
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1, 4, 9, 16, 25, 36, 49 ……….625 represents a sequence of squares of natural numbers till 25.
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3, 7, 11, 15, 19,………..87 forms another sequence, where each of the terms exceeds the preceding term by 4.
If all the terms of a progression except the first one exceed the preceding term by a fixed number, then the progression is called arithmetic progression. If a is the first term of a finite AP and d is a common difference, then AP is written as – a, a+d, a+2d, ………, a+(n-1)d.
The proof for the question can be done using the following way
- The sum of the number can be represented as
Sum = 1+2+3+4+……………+ 97 + 98 + 99 + 100——————————————– (1)
- Even if the order of the numbers is reversed, their sum remains the same.
Sum = 100 + 99 + 98 + 97 + ………..+ 4 + 3 + 2 + 1—————————————– (2)
Adding equations 1 and 2, we get
- 2 × Sum = (100+ 1) + (99+2) + (98+3 )+ (97 +4)+ ………..(4+97)+(3+98)+(2+99)+(1+100)
- 2 × Sum = 101 + 101 + 101 + 101 + ………..(4+97)+(3+98)+(2+99)+(1+100)
- 2 ×Sum = 101 (1 + 1 + 1 + …..100 terms)
- 2 × Sum = 101 (100)
- Sum = {101 × 100}/{2}
- Sum = 5050
Using the above method, the um of numbers like 1000, 10000, etc. can also be calculated.
Applications of the Sum Formula
Examination Preparation: Understanding the sum of an AP is crucial for students, especially those following the CBSE NCERT Class 10 curriculum.
Financial Analysis: In finance, the sum of an AP is employed to calculate the total value of investments or loans over a specific period.
Engineering Calculations: Engineers often use APs to model linear relationships, and the sum formula aids in analyzing the cumulative effect of these relationships.
Sum of Natural Numbers
Numbers | Sum |
1-10 | 55 |
1-100 | 5050 |
1-1000 | 500500 |
1-10000 | 50005000 |
1-100000 | 5000050000 |
1-1000000 | 500000500000 |
The interesting thing is that the above method is applicable to any AP (if the last term of the AP is known). Consider the general form of AP with the first term as a common difference as d and the last term i.e. the nth term as l. The sum of n terms of AP will be:
Sum = a + (a+d) + (a+2d) …… + (l-2d) + (l-d) + l——————– (3)
where l= a+(n-1)d
Writing in reverse order, the sum will still remain the same.
Sum =l+(l-d)+(l-2d)..…+(a+2d)+(a+d)+a——————- (4)
Solved Examples on Sum of n Terms
Example 1: If the first term of an AP is 67 and the common difference is -13, find the sum of the first 20 terms.
Solution: Here, a = 67 and d= -13
Sn = n/2[2a+(n-1)d]
S20 =20/2[2×67+(20-1)(-13)]
S20= 10[134 – 247]
S20 = -1130
So, the sum of the first 20 terms is -1130.
CBSE Class 10th Downloadable Resources:
1. CBSE Class 10th Topic Wise Summary | View Page / Download |
2. CBSE Class 10th NCERT Books | View Page / Download |
3. CBSE Class 10th NCERT Solutions | View Page / Download |
4. CBSE Class 10th Exemplar | View Page / Download |
5. CBSE Class 10th Previous Year Papers | View Page / Download |
6. CBSE Class 10th Sample Papers | View Page / Download |
7. CBSE Class 10th Question Bank | View Page / Download |
8. CBSE Class 10th Topic Wise Revision Notes | View Page / Download |
9. CBSE Class 10th Last Minutes Preparation Resources (LMP) | View Page / Download |
10. CBSE Class 10th Best Reference Books | View Page / Download |
11. CBSE Class 10th Formula Booklet | View Page / Download |
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SAMPLE PRACTICE QUESTION
Q1: What does "Sum of First n Terms" in an Arithmetic Progression (AP) signify?
Ans: The sum of the first n terms in an AP refers to the total value obtained by adding the terms from the first term to the nth term.
Q2: How is the sum of the first n terms in an AP denoted and calculated?
Ans: The sum is often denoted as \(S_n\) and calculated using the formula \(S_n = \frac{n}{2}[2a + (n-1)d]\), where \(a\) is the first term, \(n\) is the number of terms, and \(d\) is the common difference.
Q3: Can you provide a step-by-step example of finding the sum of the first n terms in an AP?
Ans: Certainly! For an AP with a first term (\(a\)) of 2, a common difference (\(d\)) of 3, and 5 terms (\(n\)), the sum formula \(S_n = \frac{5}{2}[2 \times 2 + (5-1) \times 3]\) calculates the sum.
Q4: How does the common difference impact the sum of the first n terms in an AP?**
Ans: The common difference influences the rate at which the terms increase. It is a key factor in the formula and contributes to the overall sum.
Q5: Is the formula for the sum of the first n terms applicable to APs with negative common differences?**
Ans: Yes, the formula is applicable regardless of the sign of the common difference. It accommodates both increasing and decreasing APs.