Definitions [2]
A sequence is a set of numbers arranged in a definite order.
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Each number in a sequence is called a term.
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A sequence may be written using symbols
t1,t2,t3,…,tn - The general term of a sequence is denoted by tn.
An Arithmetic Progression (A.P.) is a sequence in which the difference between consecutive terms is constant.
- Common difference = d = second term − first term
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The general form of an AP is a, a + d, a + 2d, a + 3d, …
a = first term
d = common difference
Formulae [7]
tn = a + (n − 1)d
Used when the first term a, common difference d, and term number n are known
l = a + (n − 1)d
nth term from end = l − (n − 1) d
Can be used only after finding:
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last term l
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number of terms
Common difference:
d = t2 − t1
Nature of A.P.
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d > 0→ Increasing
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d < 0→ Decreasing
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d → Constant
If a, n, l are known → \[S_n=\frac{n}{2}(a+l)\]
If a, n, d are known → \[S_n=\frac{n}{2}\left[2a+(n-1)d\right]\]
\[1+3+\cdots+(2n-1)=n^2\]
\[1+2+\cdots+n=\frac{n(n+1)}{2}\]
\[2+4+\cdots+2n=n(n+1)\]
Important Questions [23]
- Find the term t15 of an A.P. : 4, 9, 14, …………..
- Decide whether the following sequence is an A.P., if so find the 20th term of the progression: –12, –5, 2, 9, 16, 23, 30, ..............
- Find the 19th term of the following A.P.: 7, 13, 19, 25, ...
- For a given A.P. a = 3.5, d = 0, then tn = _______.
- Find the 23rd Term of the Following A.P.: 9, 4,-1,-6,-11.
- If the 9th term of an A.P. is zero, then prove that 29th term is double of 19th term.
- If the Second Term and the Fourth Term of an A.P. Are 12 and 20 Respectively, Then Find the Sum of First 25 Terms:
- The 11th Term and the 21st Term of an A.P Are 16 and 29 Respectively, Then Find the First Term, Common Difference and the 34th Term.
- Find the Sum of All Members from 50 to 250 Which Divisible by 6 and Find T13.
- Obtain the Sum of the First 56 Terms of an A.P. Whose 18th And 39th Terms Are 52 and 148 Respectively.
- Find second and third terms of an A.P. whose first term is – 2 and the common difference is – 2.
- In an A.P. sum of three consecutive terms is 27 and their products is 504. Find the terms.(Assume that three consecutive terms in an A.P. are a – d, a, a + d.)
- Complete the following activity to find the 19th term of an A.P. 7, 13, 19, 25, ........ : Activity: Given A.P. : 7, 13, 19, 25, .......... Here first term a = 7; t19 = ?
- Find the sum of first 'n' even natural numbers.
- Measures of angles of a triangle are in A.P. The measure of smallest angle is five times of common difference. Find the measures of all angles of a triangle.
- If the first term of an A.P. is p, second term is q and last term is r, then show that sum of all terms is (q+r-2p)×(p+r)2(q-p).
- Find the sum of all numbers from 50 to 350 which are divisible by 6. Hence find the 15th term of that A.P.
- In a ‘Mahila Bachat Gat’, Kavita invested from the first day of month ₹ 20 on first day, ₹ 40 on second day and ₹ 60 on third day. If she saves like this, then what would be her total savings in the
- Write an A.P. Whose First Term is a and Common Difference is D In the Following. A = 10, D = 5
- First term and the common differences of an A.P. are 6 and 3 respectively; find S27. Solution: First term = a = 6, common difference = d = 3, S27 = ?
- For an given A.P., t7 = 4, d = −4, then a = ______.
- Choose the correct alternative answer for the following question. For an given A.P. a = 3.5, d = 0, n = 101, then tn = ______.
- If the sum of first p terms of an A.P. is equal to the sum of first q terms then show that the sum of its first (p + q) terms is zero. (p ≠ q)
