Arithmetic Progression (AP) – Solved Problems
An Arithmetic Progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference (d).
Key Formulas:
Nth term: an = a + (n − 1)d
Sum of n terms: Sn = n/2 [2a + (n − 1)d]
Nth term: an = a + (n − 1)d
Sum of n terms: Sn = n/2 [2a + (n − 1)d]
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Problem 1: Find the 10th term of the AP
The first term of an AP is 3 and the common difference is 5. Find the 10th term.
Solution:
Given: a = 3, d = 5, n = 10
Using formula:
an = a + (n − 1)d
a10 = 3 + (10 − 1) × 5
a10 = 3 + 45 = 48
Given: a = 3, d = 5, n = 10
Using formula:
an = a + (n − 1)d
a10 = 3 + (10 − 1) × 5
a10 = 3 + 45 = 48
Problem 2: Find the sum of first 20 terms
Find the sum of the first 20 terms of the AP: 2, 5, 8, 11, …
Solution:
a = 2, d = 3, n = 20
Sn = n/2 [2a + (n − 1)d]
S20 = 20/2 [2×2 + 19×3]
= 10 [4 + 57] = 10 × 61
S = 610
a = 2, d = 3, n = 20
Sn = n/2 [2a + (n − 1)d]
S20 = 20/2 [2×2 + 19×3]
= 10 [4 + 57] = 10 × 61
S = 610
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Problem 3: Find the common difference
If the 7th term of an AP is 31 and the first term is 7, find the common difference.
Solution:
a = 7, a7 = 31
an = a + (n − 1)d
31 = 7 + 6d
6d = 24
d = 4
a = 7, a7 = 31
an = a + (n − 1)d
31 = 7 + 6d
6d = 24
d = 4
Problem 4: How many terms are in the AP?
Find the number of terms in the AP: 5, 9, 13, …, 77
Solution:
a = 5, d = 4, an = 77
77 = 5 + (n − 1)4
72 = 4(n − 1)
n − 1 = 18
n = 19
a = 5, d = 4, an = 77
77 = 5 + (n − 1)4
72 = 4(n − 1)
n − 1 = 18
n = 19
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Problem 5: AP Word Problem
A student saves ₹50 in the first month and increases the saving by ₹20 every month.
How much does he save in 12 months?
Solution:
a = 50, d = 20, n = 12
Sn = n/2 [2a + (n − 1)d]
S12 = 12/2 [100 + 11×20]
= 6 [100 + 220]
= 6 × 320
Total savings = ₹1920
a = 50, d = 20, n = 12
Sn = n/2 [2a + (n − 1)d]
S12 = 12/2 [100 + 11×20]
= 6 [100 + 220]
= 6 × 320
Total savings = ₹1920
Key Takeaways
- AP problems rely heavily on formulas
- Always identify a, d, and n
- Word problems usually test understanding of real-life sequences
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