Physics and Mathematics
Sum of Arithmetic Progression
1. What is Meant by Sum of First n Terms?
If an Arithmetic Progression (AP) is
[a, a + d, a + 2d, …]
then the sum of its first [n] terms means adding the first [n] consecutive terms of the AP.
This sum is denoted by [Sₙ].
2. Formula for Sum of First n Terms
(i) When first term and common difference are known
[Sₙ = \dfrac{n}{2} [2a + (n − 1)d]]
where
- [a] = first term
- [d] = common difference
- [n] = number of terms
(ii) When first term and last term are known
[Sₙ = \dfrac{n}{2} (a + l)]
where
- [l] = last term
3. Why Does This Formula Work?
The sum of an AP is obtained by:
- writing the series forward and backward,
- adding corresponding terms,
- observing that each pair gives the same sum.
This leads to the factor [n/2] naturally.
4. Important Observations
- The formula works for increasing, decreasing, constant APs
- [d] can be positive, negative, or zero
- [n] must always be a positive integer
5. Examples with Solution
Example 1
Find the sum of first 10 terms of the AP [2, 5, 8, …].
Solution:
a = 2, d = 3, n = 10
[S₁₀ = \dfrac{10}{2} [2(2) + 9 × 3]]
[S₁₀ = 5(4 + 27) = 155]
Example 2
Find the sum of first 20 terms of the AP [7, 7, 7, …].
Solution:
a = 7, d = 0, n = 20
[S₂₀ = \dfrac{20}{2} [14]]
[S₂₀ = 140]
Example 3
Find the sum of first 15 terms of the AP [−3, −7, −11, …].
Solution:
a = −3, d = −4, n = 15
[S₁₅ = \dfrac{15}{2} [−6 − 56]]
[S₁₅ = −465]
Example 4
Find the sum of first 8 terms of the AP [1/2, 1, 3/2, …].
Solution:
a = 1/2, d = 1/2, n = 8
[S₈ = \dfrac{8}{2} [1 + 7/2]]
[S₈ = 4 × 9/2 = 18]
Example 5
If the first term is 5 and the 10ᵗʰ term is 50, find the sum.
Solution:
[S₁₀ = \dfrac{10}{2} (5 + 50) = 275]
6. Conceptual Questions with Solutions
1. What does Sₙ represent in an Arithmetic Progression?
Sₙ represents the sum of the first n terms of an arithmetic progression. It does not represent a single term, but the total obtained by adding the first n consecutive terms. This notation helps us distinguish between an individual term [aₙ] and the cumulative sum [Sₙ].
2. Why do we use the symbol n in the sum formula?
The symbol n represents the number of terms being added. Since an AP can have infinitely many terms, we must clearly specify how many terms we are summing. Therefore, n must always be a positive integer.
3. Is the sum formula valid for a decreasing AP?
Yes, the formula is valid even for a decreasing AP. In a decreasing AP, the common difference d is negative, but the derivation of the formula remains unchanged. Hence, the same formula works for increasing, decreasing, and constant APs.
4. Why does the sum formula contain the factor n/2?
The factor n/2 appears because the derivation is based on pairing terms. When the first and last terms are added, the second and second-last terms are added, and so on, each pair gives the same sum. There are n such terms, so the average is multiplied by n/2.
5. Can the sum of an AP be negative?
Yes, the sum can be negative. This happens when most of the terms of the AP are negative, such as in a decreasing AP with negative first term. The sign of the sum depends on the values of a, d, and n.
6. What happens to the sum if d = 0?
If d = 0, all terms of the AP are equal. In this case, the sum becomes simply n × a. The general formula still works, but it simplifies naturally.
7. Is it necessary to know the last term to find the sum?
No, it is not necessary. If the common difference d is known, we can use the formula [Sₙ = n/2 [2a + (n − 1)d]]. The last term is only needed when d is not given.
8. Why is Sₙ a quadratic expression in n?
Because the formula contains the term (n − 1)d multiplied by n, the highest power of n becomes n². Hence, Sₙ is quadratic in n, unlike the nᵗʰ term which is linear.
9. Can we find the sum if terms are fractions?
Yes. The formula is purely algebraic and works for integers, fractions, and decimals. There is no restriction on the nature of the terms.
10. Does the order of terms affect the sum?
No. Addition is commutative, so rearranging the order of terms does not change the sum. However, the AP structure helps us apply the formula easily.
11. Can n be zero while finding the sum?
No. If n = 0, there are no terms to add. In Arithmetic Progressions, n must always be a positive integer.
12. Can the sum of first n terms be zero?
Yes. This happens when positive and negative terms cancel each other. For example, in symmetric APs around zero, the sum may become zero.
13. Is the sum always increasing as n increases?
Not always. If d ≥ 0, the sum increases with n. But if d is negative, the sum may increase initially and then decrease.
14. Why are there two formulas for the sum?
The two formulas exist for convenience. One uses a and d, and the other uses a and l. Depending on the given data, one formula may be easier to apply.
15. Why is the sum formula important?
It allows us to find the sum of many terms quickly without listing them. It is widely used in word problems, finance, physics, and statistics.
7. FAQs / Common Misconceptions
1. The sum formula works only for increasing APs.
This is incorrect. The formula works for increasing, decreasing, and constant APs because it is derived mathematically.
2. n can be any real number.
Wrong. Since we are counting terms, n must be a positive integer.
3. The last term must always be known.
False. If the common difference is known, the last term is not required.
4. Sₙ and aₙ represent the same thing.
Incorrect. aₙ is the value of the nᵗʰ term, whereas Sₙ is the sum of first n terms.
5. The formula fails when d is negative.
False. A negative d only indicates a decreasing AP; the formula still holds.
6. If terms are fractional, the sum formula cannot be used.
Incorrect. The formula works for all real numbers.
7. Constant AP always has zero sum.
Wrong. The sum equals n × a, which is zero only if a = 0.
8. The sum grows linearly with n.
Incorrect. The sum grows quadratically with n.
9. Order of writing terms affects the sum.
False. Addition does not depend on order.
10. The sum formula is assumed, not proved.
Wrong. The formula is properly derived using logical steps.
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