Arithmetic progression
The chapter 'Arithmetic progression' is a highly scoring and important topic for the board exams, because it reinforces fundamental concepts and problem-solving skills. It carries 4-6 marks in the examination. It covers concepts like, finding common difference, check whether given sequence is an A.P., problems based on \(n^{th}\) terms and sum of \(n^{th}\) terms.
Most possible variation of question types that we can expect in board exam as per previous year question paper are discussed below.
| Total Marks \(4\) - \(6\) | |
| Variation \(1\) | Variation \(2\) |
Total Mark \(= 6\)
|
Total Mark \(= 4\)
|
To prepare well for the board examination, it is necessary to understand the following concepts clearly.
- Common differences - based on some common difference or finding common difference
- Checking as an A.P - checking whether given sequence/situation forms an A.P sequence or not
- \(n^{th}\) term of an A.P - finding \(n^{th}\) term and number of terms of an A.P.
- Sum of \(n^{th}\) term of an A.P. - finding the sum of first n terms of an A.P.
| Important Concept (Learning Outcomes) | Expected Question Type | Concept dealt with |
|
Sec A -MCQ,
Sec B
|
|
|
Sec A, Sec C | |
|
Sec D, Sec E |
Let us recall the important concpets/formulas involved in Arithmetic progression:
| TO FIND | FORMULAS |
| 1. \(n^{th}\) term | \(a_{n} = a +(n-1)d\) |
| 2. No of terms |
\(n=(\frac{l-a}{d})+1\), \(l= a_{n}\)
|
| 3. Common difference | \(d= a_{2}-a_{1} = a_{3}-a_{2} = a_{4}-a_{3}=...\) |
| 4. General term of an AP | \(a, a+d, a+2d, a+3d,...\) \(a_{1}=a , a_{2} = a+d, a_{3}=a+2d, ...\) |
| 5. \(3\) consecutive terms in AP | \(a-d,a,a+d\) |
| 6. \(4\) consecutive terms in AP | \(a-3d,a-d,a+d,a+3d\) |
| 7. Sum of \(n\) terms of an AP | \(S_{n}= \frac{n}{2}(2a+(n-1)d)\) where \(n,d,a\) given |
| 8. Sum of \(n\) terms of an AP | \(S_{n}= \frac{n}{2}(a+l)\) where \(n,l,a\) given |
| 9. \(a_{n}\) if \(s_{n}\) is given |
\(a_{n} = S_{n}-S_{n-1}\)
|
Common mistakes to avoid:
1. Confusing \(n\) and \(a_n\): Remember that \(n\) represents the position of a term, while \(a_n\) represents the value of the term.
2. Incorrect sign of common difference
3. Missing brackets in the \(S_n\) formula: carefully, ensuirng the entire bracket is multiplied by \(\frac{n}{2}\)
4. Confusing \(S_n\) with \(a_n\): If \(S_n\) is given, find the term using \(a_n = S_n - S_{n-1}\) do not directly treat \(S_n\) as \(a_n\).
5. The value of \(n\) must be positive integer. Reject negative, zero or fractional values of \(n\) as they do not represent a term position.
Exam tips & tricks:
1. Use convenient forms for consecutive \(AP\) terms.
2. Choose the appropriate \(S_n\) formula to find the answer.
3. Quickly verify three terms: For \(a\), \(b\), \(c\) to be in \(AP\) chekc whether \(2b = a+c\)
4. Use \(S_n=\frac{n(n+1)}{2}\) to quickly find sums such as \(1 +2 +3 + \cdots + 100\).
5. Decode word problems first: Write the first few terms of situations such as rent, savings, salary or depreciation to identify \(a\) and \(d\) correctly before applying a formula.
6. When \(S_n\) is given algebraically, substitue \(n =1\). Since \(S_1 = a_1\), this immediately helps identify the first term and verify your expression.