C. नहीं, लगातार अंतर स्थिर नहीं है/No, the consecutive difference is not constant
Step 1
Concept
The terms are \(0,0,2,6,\ldots\), and the differences \(0,2,4,\ldots\) change. In exams, expand the product or test initial terms.
Step 2
Why this answer is correct
The correct answer is C. नहीं, लगातार अंतर स्थिर नहीं है / No, the consecutive difference is not constant. The terms are \(0,0,2,6,\ldots\), and the differences \(0,2,4,\ldots\) change. In exams, expand the product or test initial terms.
Step 3
Exam Tip
पद \(0,0,2,6,\ldots\) हैं और अंतर \(0,2,4,\ldots\) बदलते हैं। परीक्षा में गुणन रूप को फैलाकर या पहले पदों से जांचें।
D. नहीं, लगातार अंतर स्थिर नहीं है/No, the consecutive difference is not constant
Step 1
Concept
The terms are \(\frac{1}{3},\frac{1}{5},\frac{1}{7},\ldots\), and the differences change. In exams, test fractions with changing denominators by differences.
Step 2
Why this answer is correct
The correct answer is D. नहीं, लगातार अंतर स्थिर नहीं है / No, the consecutive difference is not constant. The terms are \(\frac{1}{3},\frac{1}{5},\frac{1}{7},\ldots\), and the differences change. In exams, test fractions with changing denominators by differences.
Step 3
Exam Tip
पद \(\frac{1}{3},\frac{1}{5},\frac{1}{7},\ldots\) देते हैं और अंतर बदलते हैं। परीक्षा में हर बदलने वाले भिन्नों को अंतर से परखें।
B. नहीं, लगातार अंतर बदलता है/No, the consecutive difference changes
Step 1
Concept
The initial terms are \(1,5,5,9,\ldots\), so the difference is not constant. In exams, test terms containing ((-1)^n) by substituting a few values.
Step 2
Why this answer is correct
The correct answer is B. नहीं, लगातार अंतर बदलता है / No, the consecutive difference changes. The initial terms are \(1,5,5,9,\ldots\), so the difference is not constant. In exams, test terms containing ((-1)^n) by substituting a few values.
Step 3
Exam Tip
प्रारंभिक पद \(1,5,5,9,\ldots\) देते हैं, इसलिए अंतर स्थिर नहीं है। परीक्षा में ((-1)^n) वाले पदों को कुछ मान रखकर जांचें।
C. नहीं, अंतर \(4,12,36,\ldots\) हैं/No, the differences are \(4,12,36,\ldots\)
Step 1
Concept
The consecutive differences are not equal, so it is not an AP. In exams, do not confuse a multiplicative pattern with an AP.
Step 2
Why this answer is correct
The correct answer is C. नहीं, अंतर \(4,12,36,\ldots\) हैं / No, the differences are \(4,12,36,\ldots\). The consecutive differences are not equal, so it is not an AP. In exams, do not confuse a multiplicative pattern with an AP.
Step 3
Exam Tip
लगातार अंतर समान नहीं हैं, इसलिए यह समांतर श्रेणी नहीं है। परीक्षा में गुणन पैटर्न को समांतर श्रेणी न मानें।
C. समांतर श्रेणी नहीं है क्योंकि अंतर \(2,3,4,5,\ldots\) हैं/It is not an AP because the differences are \(2,3,4,5,\ldots\)
Step 1
Concept
The consecutive differences keep increasing, so they are not constant. In exams, a famous pattern need not be an AP.
Step 2
Why this answer is correct
The correct answer is C. समांतर श्रेणी नहीं है क्योंकि अंतर \(2,3,4,5,\ldots\) हैं / It is not an AP because the differences are \(2,3,4,5,\ldots\). The consecutive differences keep increasing, so they are not constant. In exams, a famous pattern need not be an AP.
Step 3
Exam Tip
लगातार अंतर बढ़ते जा रहे हैं, इसलिए वे स्थिर नहीं हैं। परीक्षा में प्रसिद्ध पैटर्न होने से समांतर श्रेणी होना जरूरी नहीं।
D. नहीं, अंतर \(7,19,37,\ldots\) हैं/No, the differences are \(7,19,37,\ldots\)
Step 1
Concept
The consecutive differences are not equal, so it is not an AP. In exams, test constancy of differences, not just the visible pattern.
Step 2
Why this answer is correct
The correct answer is D. नहीं, अंतर \(7,19,37,\ldots\) हैं / No, the differences are \(7,19,37,\ldots\). The consecutive differences are not equal, so it is not an AP. In exams, test constancy of differences, not just the visible pattern.
Step 3
Exam Tip
लगातार अंतर बराबर नहीं हैं, इसलिए यह समांतर श्रेणी नहीं है। परीक्षा में नाम या पैटर्न नहीं, अंतर की स्थिरता जांचें।
C. नहीं, लगातार अंतर स्थिर नहीं है/No, the consecutive difference is not constant
Step 1
Concept
The first terms are \(2,\frac{3}{2},\frac{4}{3},\ldots\), and the differences change. In exams, test fractional forms in (n) by differences.
Step 2
Why this answer is correct
The correct answer is C. नहीं, लगातार अंतर स्थिर नहीं है / No, the consecutive difference is not constant. The first terms are \(2,\frac{3}{2},\frac{4}{3},\ldots\), and the differences change. In exams, test fractional forms in (n) by differences.
Step 3
Exam Tip
पहले पद \(2,\frac{3}{2},\frac{4}{3},\ldots\) देते हैं और अंतर बदलते हैं। परीक्षा में भिन्नीय (n) वाले रूप को अंतर से जांचें।
C. नहीं, लगातार अंतर स्थिर नहीं है/No, the consecutive difference is not constant
Step 1
Concept
The signs alternate and the difference does not remain the same. In exams, test alternating-sign sequences by differences only.
Step 2
Why this answer is correct
The correct answer is C. नहीं, लगातार अंतर स्थिर नहीं है / No, the consecutive difference is not constant. The signs alternate and the difference does not remain the same. In exams, test alternating-sign sequences by differences only.
Step 3
Exam Tip
पदों के चिन्ह बदलते हैं और अंतर समान नहीं रहता। परीक्षा में वैकल्पिक चिन्ह वाले अनुक्रम को अंतर से ही परखें।
C. नहीं, कोई वैध (x) नहीं/No, there is no valid (x)
Step 1
Concept
The middle-term condition gives (x(x+2)=(x+1)2), which is impossible. In exams, check validity after forming the equation.
Step 2
Why this answer is correct
The correct answer is C. नहीं, कोई वैध (x) नहीं / No, there is no valid (x). The middle-term condition gives (x(x+2)=(x+1)2), which is impossible. In exams, check validity after forming the equation.
Step 3
Exam Tip
मध्य पद की शर्त से (x(x+2)=(x+1)2) मिलता है, जो असंभव है। परीक्षा में समीकरण के बाद वैधता भी जांचें।
C. नहीं, \(a_{n+1}-a_n=2n-2\) स्थिर नहीं है/No, \(a_{n+1}-a_n=2n-2\) is not constant
Step 1
Concept
Here the difference depends on (n), so it is not constant. In exams, a quadratic \(a_n\) usually does not form an AP.
Step 2
Why this answer is correct
The correct answer is C. नहीं, \(a_{n+1}-a_n=2n-2\) स्थिर नहीं है / No, \(a_{n+1}-a_n=2n-2\) is not constant. Here the difference depends on (n), so it is not constant. In exams, a quadratic \(a_n\) usually does not form an AP.
Step 3
Exam Tip
यहां अंतर (n) पर निर्भर करता है, इसलिए स्थिर नहीं है। परीक्षा में द्विघात \(a_n\) सामान्यतः समांतर श्रेणी नहीं देता।
B. नहीं, अंतर \(3,5,7,\ldots\) हैं/No, the differences are \(3,5,7,\ldots\)
Step 1
Concept
The differences are not equal, so it is not an AP. In exams, decide by checking consecutive differences, not by looking at the terms only.
Step 2
Why this answer is correct
The correct answer is B. नहीं, अंतर \(3,5,7,\ldots\) हैं / No, the differences are \(3,5,7,\ldots\). The differences are not equal, so it is not an AP. In exams, decide by checking consecutive differences, not by looking at the terms only.
Step 3
Exam Tip
अंतर बराबर नहीं हैं, इसलिए यह समांतर श्रेणी नहीं है। परीक्षा में केवल पदों को देखकर नहीं, लगातार अंतर देखकर निर्णय लें।
C. अंकगणितीय श्रेणी नहीं क्योंकि नए अंतर समान नहीं होंगे/Not an arithmetic progression because new differences will not be equal
Step 1
Concept
The new terms are \(2,8,16,\ldots\), and the differences are \(6,8,\ldots\), which are not equal. Adding squares of term numbers does not preserve equal difference.
Step 2
Why this answer is correct
The correct answer is C. अंकगणितीय श्रेणी नहीं क्योंकि नए अंतर समान नहीं होंगे / Not an arithmetic progression because new differences will not be equal. The new terms are \(2,8,16,\ldots\), and the differences are \(6,8,\ldots\), which are not equal. Adding squares of term numbers does not preserve equal difference.
Step 3
Exam Tip
नए पद \(2,8,16,\ldots\) मिलते हैं और अंतर \(6,8,\ldots\) हैं, जो समान नहीं हैं। पद संख्या के वर्ग जोड़ने से समान अंतर नहीं बचता।
C. नहीं, क्योंकि क्रमागत अंतर समान नहीं हैं/No because consecutive differences are not equal
Step 1
Concept
The new terms are (16,81,196), and the differences are (65,115), which are not equal. Squaring generally does not preserve an arithmetic progression.
Step 2
Why this answer is correct
The correct answer is C. नहीं, क्योंकि क्रमागत अंतर समान नहीं हैं / No because consecutive differences are not equal. The new terms are (16,81,196), and the differences are (65,115), which are not equal. Squaring generally does not preserve an arithmetic progression.
Step 3
Exam Tip
नए पद (16,81,196) हैं और अंतर (65,115) हैं, जो समान नहीं हैं। वर्ग करने से सामान्यतः अंकगणितीय श्रेणी सुरक्षित नहीं रहती।
C. नहीं, क्योंकि नए अंतर समान नहीं हैं/No, because new differences are not equal
Step 1
Concept
The new terms are \(1,8,21,\ldots\), and the differences are \(7,13,\ldots\), which are not equal. Multiplying by term number does not keep (d) constant.
Step 2
Why this answer is correct
The correct answer is C. नहीं, क्योंकि नए अंतर समान नहीं हैं / No, because new differences are not equal. The new terms are \(1,8,21,\ldots\), and the differences are \(7,13,\ldots\), which are not equal. Multiplying by term number does not keep (d) constant.
Step 3
Exam Tip
नए पद \(1,8,21,\ldots\) हैं और अंतर \(7,13,\ldots\) हैं, जो समान नहीं हैं। पद संख्या से गुणा करना (d) को स्थिर नहीं रखता।
C. नहीं, क्योंकि नए अंतर समान नहीं होंगे/No, because new differences will not be equal
Step 1
Concept
The added squares \(1,4,9,\ldots\) have differences \(3,5,\ldots\), which are not equal, so the new sequence will not remain arithmetic. Squares of term numbers do not create equal differences.
Step 2
Why this answer is correct
The correct answer is C. नहीं, क्योंकि नए अंतर समान नहीं होंगे / No, because new differences will not be equal. The added squares \(1,4,9,\ldots\) have differences \(3,5,\ldots\), which are not equal, so the new sequence will not remain arithmetic. Squares of term numbers do not create equal differences.
Step 3
Exam Tip
नए जोड़े गए वर्ग \(1,4,9,\ldots\) के अंतर \(3,5,\ldots\) समान नहीं हैं, इसलिए नया अनुक्रम अंकगणितीय नहीं रहेगा। पद संख्या के वर्ग से समान अंतर नहीं बनता।
C. अंकगणितीय श्रेणी नहीं क्योंकि अंतर समान नहीं हैं/Not an arithmetic progression because differences are not equal
Step 1
Concept
The new terms are (9,49,121), and the differences are (40,72), which are not equal. Squaring generally does not preserve an arithmetic progression.
Step 2
Why this answer is correct
The correct answer is C. अंकगणितीय श्रेणी नहीं क्योंकि अंतर समान नहीं हैं / Not an arithmetic progression because differences are not equal. The new terms are (9,49,121), and the differences are (40,72), which are not equal. Squaring generally does not preserve an arithmetic progression.
Step 3
Exam Tip
नए पद (9,49,121) हैं और अंतर (40,72) हैं, जो समान नहीं हैं। वर्ग करने से सामान्यतः अंकगणितीय श्रेणी सुरक्षित नहीं रहती।
The first two differences are (5,5), but (20-14=6). It is necessary to check all available consecutive differences.
Step 2
Why this answer is correct
The correct answer is A. \(4, 9, 14, 20,\ldots\). The first two differences are (5,5), but (20-14=6). It is necessary to check all available consecutive differences.
Step 3
Exam Tip
पहले दो अंतर (5,5) हैं, पर (20-14=6) है। सभी उपलब्ध क्रमागत अंतर जांचना जरूरी है।
C. नहीं, क्योंकि क्रमागत अंतर समान नहीं हैं/No because consecutive differences are not equal
Step 1
Concept
The new terms are (81,225,441), and the differences are (144,216), which are not equal. Squaring generally does not preserve an arithmetic progression.
Step 2
Why this answer is correct
The correct answer is C. नहीं, क्योंकि क्रमागत अंतर समान नहीं हैं / No because consecutive differences are not equal. The new terms are (81,225,441), and the differences are (144,216), which are not equal. Squaring generally does not preserve an arithmetic progression.
Step 3
Exam Tip
नए पद (81,225,441) हैं और अंतर (144,216) हैं, जो समान नहीं हैं। वर्ग करने से सामान्यतः अंकगणितीय श्रेणी सुरक्षित नहीं रहती।
Up to (2,6,10), the differences are (4,4), but (15-10=5). It is necessary to check all available consecutive differences.
Step 2
Why this answer is correct
The correct answer is A. \(2, 6, 10, 15,\ldots\). Up to (2,6,10), the differences are (4,4), but (15-10=5). It is necessary to check all available consecutive differences.
Step 3
Exam Tip
(2,6,10) तक अंतर (4,4) हैं, पर (15-10=5) है। सभी उपलब्ध क्रमागत अंतर जांचना जरूरी है।
C. क्योंकि क्रमागत अंतर समान नहीं हैं/Because consecutive differences are not equal
Step 1
Concept
The differences are (4,6,8), which are not equal. Do not assume arithmetic progression only because terms increase.
Step 2
Why this answer is correct
The correct answer is C. क्योंकि क्रमागत अंतर समान नहीं हैं / Because consecutive differences are not equal. The differences are (4,6,8), which are not equal. Do not assume arithmetic progression only because terms increase.
Step 3
Exam Tip
अंतर (4,6,8) हैं जो समान नहीं हैं। केवल बढ़ते पद देखकर अंकगणितीय श्रेणी न मानें।
D. क्योंकि क्रमागत अंतर समान नहीं हैं/Because consecutive differences are not equal
Step 1
Concept
The differences are (4,6,8), which are not equal. Increasing terms alone are not enough for an arithmetic progression.
Step 2
Why this answer is correct
The correct answer is D. क्योंकि क्रमागत अंतर समान नहीं हैं / Because consecutive differences are not equal. The differences are (4,6,8), which are not equal. Increasing terms alone are not enough for an arithmetic progression.
Step 3
Exam Tip
अंतर (4,6,8) हैं जो समान नहीं हैं। बढ़ते पद होना ही अंकगणितीय श्रेणी के लिए पर्याप्त नहीं है।
In \(2,5,9,14,\ldots\), the differences are (3,4,5). If differences are not equal, the sequence is not an arithmetic progression.
Step 2
Why this answer is correct
The correct answer is D. \(2,5,9,14,\ldots\). In \(2,5,9,14,\ldots\), the differences are (3,4,5). If differences are not equal, the sequence is not an arithmetic progression.
Step 3
Exam Tip
\(2,5,9,14,\ldots\) में अंतर (3,4,5) हैं। अंतर समान न हो तो अनुक्रम समांतर श्रेढ़ी नहीं होता।
