✦ NCF 2023 ALIGNED • CBSE CLASS X ✦
Chapter 5: Arithmetic Progressions
Assertion-Reason Question Bank
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Assertion (A)
The sum of the first \(10\) terms of the AP \(2, 7, 12, \dots\) is \(245\).
Reason (R)
The sum of first \(n\) terms is \(S_n = \frac{n}{2}[2a + (n-1)d] = \frac{10}{2}[2(2) + 9(5)] = 5[4 + 45] = 5 \times 49 = 245\).
Correct Option: (A)
Formula \(S_n = (n/2)[2a + (n-1)d]\) with \(a=2, d=5, n=10\) gives \(5(49) = 245\). Reason explains Assertion.
Formula \(S_n = (n/2)[2a + (n-1)d]\) with \(a=2, d=5, n=10\) gives \(5(49) = 245\). Reason explains Assertion.
Correct Answer: (A)
\(S_{10} = 245\). Reason explains Assertion.
\(S_{10} = 245\). Reason explains Assertion.
Assertion (A)
The \(n^{\text{th}}\) term of an AP is \(a_n = a + nd\).
Reason (R)
The correct formula for \(n^{\text{th}}\) term is \(a_n = a + (n-1)d\).
Correct Option: (D)
Assertion (A) is false (\(a + nd\) misses \(-1\)). Reason (R) is true.
Assertion (A) is false (\(a + nd\) misses \(-1\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The sum of first \(n\) odd natural numbers is \(n^2\).
Reason (R)
The first odd natural number is \(1\) and common difference is \(1\).
Correct Option: (C)
Assertion (A) is true (\(S_n = (n/2)[2(1) + (n-1)2] = n^2\)). Reason (R) is false because common difference of odd numbers (\(1, 3, 5, \dots\)) is \(2\), not \(1\).
Assertion (A) is true (\(S_n = (n/2)[2(1) + (n-1)2] = n^2\)). Reason (R) is false because common difference of odd numbers (\(1, 3, 5, \dots\)) is \(2\), not \(1\).
Correct Answer: (C)
Assertion true; Reason false (\(d = 2\)).
Assertion true; Reason false (\(d = 2\)).
Assertion (A)
The sequence \(\sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32}, \dots\) forms an AP.
Reason (R)
Simplifying terms gives \(\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}, \dots\) with common difference \(d = \sqrt{2}\).
Correct Option: (B)
Both A and R are true. R shows simplification of surds.
Both A and R are true. R shows simplification of surds.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
The sum of first \(n\) natural numbers is \(n^2\).
Reason (R)
The sum of first \(n\) natural numbers is given by \(\frac{n(n+1)}{2}\).
Correct Option: (D)
Assertion (A) is false (\(n^2\) is sum of odd numbers). Reason (R) is true.
Assertion (A) is false (\(n^2\) is sum of odd numbers). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The common difference of the AP \(3, 1, -1, -3, \dots\) is \(-2\).
Reason (R)
In an AP, common difference \(d = a_2 - a_1 = 1 - 3 = -2\).
Correct Option: (A)
\(d = a_2 - a_1 = 1 - 3 = -2\). Reason gives the general formula and explains Assertion.
\(d = a_2 - a_1 = 1 - 3 = -2\). Reason gives the general formula and explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If first term is \(3\) and common difference is \(0\), the terms are \(3, 0, 0, 0, \dots\).
Reason (R)
When \(d = 0\), all terms of the AP are equal to the first term (\(3, 3, 3, 3, \dots\)).
Correct Option: (D)
Assertion (A) is false. Reason (R) is true.
Assertion (A) is false. Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
If \(S_n = 3n^2 + 5n\), then \(a_n = 6n + 2\).
Reason (R)
\(a_n = S_n - S_{n-1} = (3n^2 + 5n) - [3(n-1)^2 + 5(n-1)] = 6n + 2\).
Correct Option: (B)
Both A and R are true algebra calculations.
Both A and R are true algebra calculations.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
The \(10^{\text{th}}\) term of AP \(5, 8, 11, \dots\) is \(32\).
Reason (R)
The sum of first \(10\) terms is given by \(S_{10} = 10 \times a_{10}\).
Correct Option: (C)
Assertion (A) is true (\(a_{10} = 5 + 9(3) = 32\)). Reason (R) is false because \(S_n = (n/2)(a + a_n)\), which is \(5 \times (a + a_{10})\), not \(10 \times a_{10}\).
Assertion (A) is true (\(a_{10} = 5 + 9(3) = 32\)). Reason (R) is false because \(S_n = (n/2)(a + a_n)\), which is \(5 \times (a + a_{10})\), not \(10 \times a_{10}\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
The sum of first \(n\) terms of an AP with \(a = 1, l = 11, S_n = 36\) has \(n = 6\).
Reason (R)
Sum formula using first and last terms is \(S_n = \frac{n}{2}(a + l)\).
Correct Option: (B)
Both A and R are true (\(36 = (n/2)(1 + 11) \implies 36 = 6n \implies n = 6\)).
Both A and R are true (\(36 = (n/2)(1 + 11) \implies 36 = 6n \implies n = 6\)).
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
The \(10^{\text{th}}\) term of the AP \(2, 7, 12, \dots\) is \(47\).
Reason (R)
The \(n^{\text{th}}\) term of an AP with first term \(a\) and common difference \(d\) is \(a_n = a + (n-1)d = 2 + (10-1)5 = 2 + 45 = 47\).
Correct Option: (A)
\(a = 2, d = 5\). \(a_{10} = 2 + 9(5) = 47\). Reason provides the exact formula and step-by-step substitution, explaining Assertion.
\(a = 2, d = 5\). \(a_{10} = 2 + 9(5) = 47\). Reason provides the exact formula and step-by-step substitution, explaining Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The \(0^{\text{th}}\) term of an AP exists and is equal to \(a - d\).
Reason (R)
In an AP, the term index \(n\) must always be a positive integer (\(n \in \mathbb{N}\)).
Correct Option: (D)
Assertion (A) is false (\(n\) must be positive integer \(1, 2, 3, \dots\)). Reason (R) is true.
Assertion (A) is false (\(n\) must be positive integer \(1, 2, 3, \dots\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
If \(a, b, c\) are in AP, then \(2b = a + c\).
Reason (R)
In an AP, \(b - a = c - b \implies 2b = a + c\).
Correct Option: (B)
Both A and R are true mathematical proofs.
Both A and R are true mathematical proofs.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
The common difference of an AP can be positive, negative, or zero.
Reason (R)
In the constant AP \(5, 5, 5, 5, \dots\), the common difference is \(0\).
Correct Option: (B)
Both A and R are true. R is an example of zero common difference.
Both A and R are true. R is an example of zero common difference.
Correct Answer: (B)
Both true, R provides an example.
Both true, R provides an example.
Assertion (A)
The \(n^{\text{th}}\) term of the AP \(1/m, (1+m)/m, (1+2m)/m, \dots\) is \(1/m + n - 1\).
Reason (R)
The common difference of this AP is \(1/m\).
Correct Option: (C)
Assertion (A) is true (\(a = 1/m, d = 1\). \(a_n = 1/m + (n-1)1 = 1/m + n - 1\)). Reason (R) is false because \(d = (1+m)/m - 1/m = m/m = 1\), not \(1/m\).
Assertion (A) is true (\(a = 1/m, d = 1\). \(a_n = 1/m + (n-1)1 = 1/m + n - 1\)). Reason (R) is false because \(d = (1+m)/m - 1/m = m/m = 1\), not \(1/m\).
Correct Answer: (C)
Assertion true; Reason false (\(d=1\)).
Assertion true; Reason false (\(d=1\)).
Assertion (A)
The sum of the first \(n\) positive integers is given by \(S_n = \frac{n(n+1)}{2}\).
Reason (R)
Applying \(S_n = \frac{n}{2}(a + l)\) with first term \(a = 1\) and last term \(l = n\) gives \(\frac{n}{2}(1 + n) = \frac{n(n+1)}{2}\).
Correct Option: (A)
Using formula \(S_n = (n/2)(a+l)\) with \(a=1, l=n\) proves \(S_n = n(n+1)/2\). Reason explains Assertion.
Using formula \(S_n = (n/2)(a+l)\) with \(a=1, l=n\) proves \(S_n = n(n+1)/2\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If \(a_n = 2n + 1\), the common difference \(d = 1\).
Reason (R)
\(a_1 = 3, a_2 = 5 \implies d = a_2 - a_1 = 5 - 3 = 2\).
Correct Option: (D)
Assertion (A) is false (\(d = 2\), not \(1\)). Reason (R) is true.
Assertion (A) is false (\(d = 2\), not \(1\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The \(11^{\text{th}}\) term from the end of the AP \(10, 7, 4, \dots, -62\) is \(-32\).
Reason (R)
The \(n^{\text{th}}\) term from the end is given by \(l - (n-1)d = -62 - (11-1)(-3) = -62 + 30 = -32\).
Correct Option: (A)
Last term \(l = -62, d = -3\). \(11^{\text{th}}\) term from end \(= -62 - 10(-3) = -32\). Reason explains Assertion.
Last term \(l = -62, d = -3\). \(11^{\text{th}}\) term from end \(= -62 - 10(-3) = -32\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The \(10^{\text{th}}\) term of AP \(10, 6, 2, -2, \dots\) is \(26\).
Reason (R)
\(a_{10} = a + 9d = 10 + 9(-4) = 10 - 36 = -26\).
Correct Option: (D)
Assertion (A) is false (it is \(-26\), not \(26\)). Reason (R) is true.
Assertion (A) is false (it is \(-26\), not \(26\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
If the \(n^{\text{th}}\) term of an AP is \(a_n = 3 + 4n\), then its common difference is \(4\).
Reason (R)
\(a_1 = 7, a_2 = 11 \implies d = a_2 - a_1 = 11 - 7 = 4\).
Correct Option: (A)
\(a_n = 3 + 4n \implies a_1 = 7, a_2 = 11, d = 4\). Reason explains Assertion.
\(a_n = 3 + 4n \implies a_1 = 7, a_2 = 11, d = 4\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The sum of first \(20\) terms of AP \(5, 8, 11, 14, \dots\) is \(670\).
Reason (R)
Formula for sum of \(n\) terms is \(S_n = \frac{n}{2}[a + (n-1)d]\).
Correct Option: (C)
Assertion (A) is true (\(S_{20} = 10[2(5) + 19(3)] = 10[10 + 57] = 670\)). Reason (R) is false because formula missing factor \(2\) in \(2a\) (\(2a\), not \(a\)).
Assertion (A) is true (\(S_{20} = 10[2(5) + 19(3)] = 10[10 + 57] = 670\)). Reason (R) is false because formula missing factor \(2\) in \(2a\) (\(2a\), not \(a\)).
Correct Answer: (C)
Assertion true; Reason false (formula should be \([2a + (n-1)d]\)).
Assertion true; Reason false (formula should be \([2a + (n-1)d]\)).
Assertion (A)
In an AP, if \(d = -4, n = 7, a_n = 4\), then first term \(a = 28\).
Reason (R)
The \(n^{\text{th}}\) term formula is \(a_n = a + (n-1)d\).
Correct Option: (B)
Both statements are true. \(4 = a + 6(-4) \implies 4 = a - 24 \implies a = 28\). R is the formula.
Both statements are true. \(4 = a + 6(-4) \implies 4 = a - 24 \implies a = 28\). R is the formula.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
The sequence \(a, a+d, a+2d, \dots\) has \(a_5 = a + 4d\).
Reason (R)
The common difference of AP \(a, a+d, a+2d, \dots\) is \(a\).
Correct Option: (C)
Assertion (A) is true. Reason (R) is false because common difference is \(d\), not \(a\).
Assertion (A) is true. Reason (R) is false because common difference is \(d\), not \(a\).
Correct Answer: (C)
Assertion true; Reason false (\(d\) is common difference).
Assertion true; Reason false (\(d\) is common difference).
Assertion (A)
The \(15^{\text{th}}\) term of \(a, a+d, a+2d, \dots\) is \(a + 14d\).
Reason (R)
The first term of an AP is denoted by \(a\).
Correct Option: (B)
Both are true statements, but R is just notation, not full formula derivation.
Both are true statements, but R is just notation, not full formula derivation.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
The sum of \(n\) terms of an AP with \(a = 2, d = 2\) is \(2n^2\).
Reason (R)
\(S_n = \frac{n}{2}[2(2) + (n-1)2] = \frac{n}{2}[4 + 2n - 2] = \frac{n}{2}[2n + 2] = n(n+1) = n^2 + n\).
Correct Option: (D)
Assertion (A) is false (\(S_n = n^2 + n\), not \(2n^2\)). Reason (R) is true.
Assertion (A) is false (\(S_n = n^2 + n\), not \(2n^2\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
If \(2x, x+10, 3x+2\) are in AP, then \(x = 6\).
Reason (R)
If three terms are in AP, \(a+b=c\).
Correct Option: (C)
Assertion (A) is true (\(2(x+10) = 2x + 3x + 2 \implies 2x+20 = 5x+2 \implies 3x = 18 \implies x = 6\)). Reason (R) is false because condition is \(2b = a + c\), not \(a+b=c\).
Assertion (A) is true (\(2(x+10) = 2x + 3x + 2 \implies 2x+20 = 5x+2 \implies 3x = 18 \implies x = 6\)). Reason (R) is false because condition is \(2b = a + c\), not \(a+b=c\).
Correct Answer: (C)
Assertion true; Reason false (\(2b = a+c\)).
Assertion true; Reason false (\(2b = a+c\)).
Assertion (A)
The list of numbers \(2, 4, 8, 16, \dots\) is NOT an AP.
Reason (R)
In an AP, the difference between consecutive terms must be constant, but here \(4 - 2 = 2 \ne 8 - 4 = 4\).
Correct Option: (A)
Since \(a_2 - a_1 = 2\) and \(a_3 - a_2 = 4\), common difference is not constant, so it is a geometric progression, not an AP. Reason explains Assertion.
Since \(a_2 - a_1 = 2\) and \(a_3 - a_2 = 4\), common difference is not constant, so it is a geometric progression, not an AP. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The numbers \(1, 2, 4, 8, 16\) form an AP.
Reason (R)
In an AP, the difference between any two consecutive terms must be constant.
Correct Option: (D)
Assertion (A) is false (\(2-1=1 \ne 4-2=2\)). Reason (R) is true.
Assertion (A) is false (\(2-1=1 \ne 4-2=2\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The \(4^{\text{th}}\) term from the end of the AP \(-11, -8, -5, \dots, 49\) is \(40\).
Reason (R)
The common difference \(d\) of AP \(-11, -8, -5, \dots\) is \(-3\).
Correct Option: (C)
Assertion (A) is true (\(l - (4-1)d = 49 - 3(3) = 40\)). Reason (R) is false because \(d = -8 - (-11) = +3\), not \(-3\).
Assertion (A) is true (\(l - (4-1)d = 49 - 3(3) = 40\)). Reason (R) is false because \(d = -8 - (-11) = +3\), not \(-3\).
Correct Answer: (C)
Assertion true; Reason false (\(d = +3\)).
Assertion true; Reason false (\(d = +3\)).
Assertion (A)
The \(n^{\text{th}}\) term of an AP is always a linear expression in \(n\).
Reason (R)
The sum of first \(n\) terms of an AP is always a quadratic expression in \(n\) without a constant term.
Correct Option: (B)
Both A and R are true properties of AP.
Both A and R are true properties of AP.
Correct Answer: (B)
Both true.
Both true.