An arithmetic progression, also known as an arithmetic sequence, is a sequence of numbers in which the difference between any two consecutive members remains constant.
First term is denoted by a Common difference is denoted by d n^{th} term is denoted by
a_{n} or t_{n}
Sum of First n terms is denoted by S_{n}
Example : 4,8,12,16……..
nth term of an AP
Formula to find the nth term of an AP is
T_{n} = a + (n – 1) d
where t_{n} = nth term,
a = first term ,
d = common difference,
n = number of terms in the sequence.
Formula to find the numbers of term of an AP is
where
n = number of terms,
a = first term,
l = last term,
d= common difference.
Formula to find the sum of first n terms of an AP is
or
where,
a = first term,
d = common difference,
t_{n} = n^{th} term = a + (n1)d
If a, b, c are in AP, then the Arithmetic mean of a and c is b i.e.
Formula to find the sum of squares of first n natural numbers is
where
S = Sum of Squares of first n natural numbers
n = number of First n natural numbers.
Q1: The sum of the first 20 terms of an arithmetic progression is 610, and the sum of the next 30 terms is 2130. What is the common difference of this arithmetic progression?
(a) 8
(b) 10
(c) 12
(d) 15
Ans: (b)
Let the first term of the arithmetic progression be ‘a’ and the common difference be ‘d’. The sum of the first 20 terms is given by:
The sum of the next 30 terms is:
Given S_{1} = 610 and S₂ = 2130, we can write two equations:
20a + 190d = 610
30a + 590d = 2130
Solving these equations, we get d = 10. Therefore, the correct answer is (b) 10.
Q2: The sum of the first 50 terms of an arithmetic progression is 4800. If the first term is 12 and the common difference is 3, find the 30th term.
(a) 14
(b) 17
(c) 20
(d) 75
Ans: (d)
The sum of the first 50 terms of the arithmetic progression is given by:
Solving, we get S_{50} = 4800.
Now, to find the 30th term, we can use the formula for the nth term of an AP:
a_{30} = a + (n1) * d
a_{30} = 12 + (301) * (3)
a_{30} = 12 – 87
a_{30} = 75.
The correct answer is 75.
Q3: The 10th term of an arithmetic progression is equal to three times the 6th term. If the sum of the first 10 terms is 220, what is the common difference of the progression?
(a) 6
(b) 8
(c) 10
(d) 12
Ans: (b)
Let the first term of the arithmetic progression be ‘a’, and the common difference be ‘d’.
According to the problem, a + 9d = 3(a + 5d)
Solving this equation, we get a = 10d.
Now, the sum of the first 10 terms of the arithmetic progression is given by:
Solving this equation, we get d = 8.
Therefore, the correct answer is (B) 8.
Q4: In an arithmetic progression, the 15th term is 8 more than the 8th term. If the common difference is 3, what is the 10th term of the progression?
(a) 21
(b) 23
(c) 25
(d) 40
Ans: (d)
Let the first term of the arithmetic progression be ‘a’. Then, the 8th term is given by a + 7 * 3 = a + 21, and the 15th term is a + 14 * 3 = a + 42.
According to the problem, a + 42 = (a + 21) + 8
Solving this equation, we get a = 13.
Now, the 10th term is a + 9 * 3 = 13 + 27 = 40.
The correct answer is 40.
Q5: If the sum of the first ‘n’ terms of an arithmetic progression is given by S_{n} = 5n_{2} + 3_{n}, what is the first term of the progression?
(a) 2
(b) 5
(c) 7
(d) 4
Ans: (d)
The sum of the first ‘n’ terms of an arithmetic progression is given by:
Comparing it with Sn = 5n^{2} + 3n, we get 2a + (n1) * d = 5n + 3
Since we are looking for the first term ‘a’, we can consider ‘n’ as 1:
2a + (11) * d = 5 + 3
2a = 8
a = 4.
The correct answer is 4.
314 videos170 docs185 tests

1. What is an arithmetic progression? 
2. How can I find the nth term of an arithmetic progression? 
3. Can an arithmetic progression have a negative common difference? 
4. What is the formula for finding the sum of an arithmetic progression? 
5. How can I determine if a given sequence is an arithmetic progression? 
314 videos170 docs185 tests


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