sequences se related questions ko ek jagah revise karein. Har question me bilingual content, answer feedback aur explanation available hai.
3683 questions tagged with sequences.
For which (n) will \(S_n-S_{n-3}=144\)?
D. (49)
(S_n-S_{n-3}=(n-2)+(n-1)+n=3n-3) है, इसलिए (3n-3=144) से (n=49)। अंतर को अंतिम तीन पदों के रूप में लिखें। / (S_n-S_{n-3}=(n-2)+(n-1)+n=3n-3), so (3n-3=144) gives (n=49). Write the difference as the last three terms.
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What is the ratio of \(S_{60}\) to \(S_{30}\)?
A. (122:31)
\(S_{60}:S_{30}=1830:465=122:31\) है। अनुपात को सबसे बड़े समान गुणक से सरल करें। / \(S_{60}:S_{30}=1830:465=122:31\). Simplify the ratio by the greatest common factor.
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If \(1+2+3+\cdots+n=2556\), which is (n)?
C. (71)
\(S_{71}=\frac{71\times72}{2}=2556\) इसलिए (n=71) है। विकल्पों को सूत्र में रखकर तेज जांच करें। / \(S_{71}=\frac{71\times72}{2}=2556\), so (n=71). Substitute options in the formula for a quick check.
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C. (610)
अंतर \(S_{40}-S_{20}=820-210=610\) है। तुलना में दोनों योगों को अलग-अलग निकालें। / The difference is \(S_{40}-S_{20}=820-210=610\). In comparison questions, find both sums separately.
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What is the value of \(S_{49}+S_{51}-2S_{50}\)?
B. (1)
\(S_{51}=S_{50}+51\) और \(S_{49}=S_{50}-50\), इसलिए मान (1) है। लगातार योगों के अंतर का उपयोग करें। / \(S_{51}=S_{50}+51\) and \(S_{49}=S_{50}-50\), so the value is (1). Use differences of consecutive sums.
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If \(S_n=465\), what will be the value of \(S_{n+5}\)?
A. (630)
\(S_{30}=465\) इसलिए \(S_{35}=630\) होगा। पहले (n) पहचानें फिर नया सूचकांक लगाएं। / \(S_{30}=465\), so \(S_{35}=630\). Identify (n) first and then use the new index.
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What will be the value of \(S_{85}-S_{80}\)?
C. (415)
\(S_{85}-S_{80}=81+82+83+84+85=415\) है। पास के योगों का अंतर बीच के पदों का योग होता है। / \(S_{85}-S_{80}=81+82+83+84+85=415\). The difference of nearby sums equals the sum of terms in between.
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What is the sum of natural numbers from (31) to (72)?
C. (2163)
योग \(S_{72}-S_{30}=2628-465=2163\) है। बीच की संख्याओं का योग निकालने के लिए पहले वाले पद घटाएं। / The sum is \(S_{72}-S_{30}=2628-465=2163\). To find a middle range sum, subtract the previous terms.
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If the sum of the first (n) natural numbers is (2016), what is the value of (n)?
C. (63)
\(S_{63}=\frac{63\times64}{2}=2016\) इसलिए (n=63) है। \(2S_n\) को दो क्रमागत संख्याओं के गुणनफल से मिलाएं। / \(S_{63}=\frac{63\times64}{2}=2016\), so (n=63). Match \(2S_n\) with the product of two consecutive numbers.
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What is the sum of the first (64) natural numbers?
B. (2080)
\(S_{64}=\frac{64\times65}{2}=2080\) है। बड़े (n) में पहले (2) से भाग देकर गणना करें। / \(S_{64}=\frac{64\times65}{2}=2080\). For large (n), divide by (2) first.
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If \(S_n=5151\), what will be the value of (n)?
B. (101)
\(\frac{101\cdot102}{2}=5151\), इसलिए (n=101)। सूत्र में विकल्प रखकर उत्तर जांचें। / \(\frac{101\cdot102}{2}=5151\), so (n=101). Check the answer by substituting options in the formula.
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What is (25%) of the sum of the first (56) natural numbers?
A. (399)
\(S_{56}=1596\) और \(25%=\frac{1}{4}\), इसलिए मान (399) है। प्रतिशत को भिन्न में बदलें। / \(S_{56}=1596\) and \(25%=\frac{1}{4}\), so the value is (399). Convert percentage into a fraction.
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If \(S_n-S_{n-1}=84\), what will be \(S_{n-1}\)?
B. (3486)
\(S_n-S_{n-1}=n\), इसलिए (n=84) और \(S_{83}=\frac{83\cdot84}{2}=3486\)। पिछले योग के लिए (n-1) लें। / \(S_n-S_{n-1}=n\), so (n=84) and \(S_{83}=\frac{83\cdot84}{2}=3486\). For the previous sum, take (n-1).
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If \(S_n=1953\), what is the value of \(S_{n+2}\)?
B. (2080)
\(1953=S_{62}\), इसलिए \(S_{64}=\frac{64\cdot65}{2}=2080\)। पहले (n) पहचानें फिर (2) आगे बढ़ें। / \(1953=S_{62}\), so \(S_{64}=\frac{64\cdot65}{2}=2080\). First identify (n) and then move (2) ahead.
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What is the difference between the sum from (1) to (150) and the sum from (1) to (100)?
B. (6275)
\(S_{150}-S_{100}=11325-5050=6275\)। यह (101) से (150) तक का योग है। / \(S_{150}-S_{100}=11325-5050=6275\). This is the sum from (101) to (150).
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If \(S_n=2346\), what is the value of (n)?
B. (68)
\(\frac{68\cdot69}{2}=2346\), इसलिए (n=68)। विकल्पों को सूत्र में रखकर सत्यापित करें। / \(\frac{68\cdot69}{2}=2346\), so (n=68). Verify the options by substituting them into the formula.
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B. (100)
\(S_{100}-S_{99}=100\) होता है। क्रमिक योगों का अंतर अगला पद होता है। / \(S_{100}-S_{99}=100\). The difference of consecutive sums is the next term.
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If \(S_{n+1}-S_{n-2}=150\), what will be the value of (n)?
B. (50)
(S_{n+1}-S_{n-2}=(n-1)+n+(n+1)=3n)। (3n=150) से (n=50)। / (S_{n+1}-S_{n-2}=(n-1)+n+(n+1)=3n). From (3n=150), (n=50).
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If \(S_n+S_{n-1}=1561\) and \(S_n-S_{n-1}=40\), which is the value of \(S_n\)?
B. (800.5)
दोनों समीकरण जोड़ने पर \(2S_n=1601\), इसलिए \(S_n=800.5\) है। यदि पूर्णांक न मिले तो दिए मानों की संगति जांचें। / Adding the two equations gives \(2S_n=1601\), so \(S_n=800.5\). If an integer is not obtained, check the consistency of given values.
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Find the sum of natural numbers from (121) to (150).
B. (4065)
योग \(S_{150}-S_{120}=11325-7260=4065\) है। \(S_b-S_{a-1}\) विधि उपयोग करें। / The sum is \(S_{150}-S_{120}=11325-7260=4065\). Use the \(S_b-S_{a-1}\) method.
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If \(S_n=4560\), what is the value of \(S_{n+5}-S_n\)?
C. (490)
\(4560=S_{95}\), इसलिए अंतर (96+97+98+99+100=490) है। अगले पांच पदों का योग लें। / \(4560=S_{95}\), so the difference is (96+97+98+99+100=490). Take the sum of the next five terms.
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What is half of the sum of the first (84) natural numbers?
B. (1785)
\(S_{84}=\frac{84\cdot85}{2}=3570\), इसलिए आधा (1785) है। पहले पूर्ण योग निकालें। / \(S_{84}=\frac{84\cdot85}{2}=3570\), so half is (1785). Find the full sum first.
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If \(S_n-S_{n-3}=180\), what is (n)?
C. (61)
\(S_n-S_{n-3}=3n-3\)। (3n-3=180) से (n=61)। / \(S_n-S_{n-3}=3n-3\). From (3n-3=180), (n=61).
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What is the sum of natural numbers from (25) to (75)?
A. (2550)
योग \(S_{75}-S_{24}=2850-300=2550\) है। निचली सीमा से एक कम तक का योग घटाएं। / The sum is \(S_{75}-S_{24}=2850-300=2550\). Subtract the sum up to one less than the lower limit.
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If \(1+2+3+\cdots+n=3160\), what is the value of (n)?
C. (79)
\(\frac{79\cdot80}{2}=3160\), इसलिए (n=79)। उत्तर को सूत्र से जांचना जरूरी है। / \(\frac{79\cdot80}{2}=3160\), so (n=79). It is important to check the answer with the formula.
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If \(S_n=3655\), what will be \(S_n-S_{n-4}\)?
C. (334)
\(3655=S_{85}\), इसलिए अंतर (82+83+84+85=334) है। अंतिम चार पदों को जोड़ें। / \(3655=S_{85}\), so the difference is (82+83+84+85=334). Add the last four terms.
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What is the ratio of the sums of the first (90) and first (60) natural numbers?
A. (91:61)
\(S_{90}:S_{60}=4095:1830=91:61\)। साझा गुणनखंड से अनुपात सरल करें। / \(S_{90}:S_{60}=4095:1830=91:61\). Simplify the ratio using the common factor.
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B. (4020)
परिणाम \(S_{120}-S_{80}=7260-3240=4020\) है। यह (81) से (120) तक का योग है। / The result is \(S_{120}-S_{80}=7260-3240=4020\). This is the sum from (81) to (120).
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If \(S_{n+2}-S_n=155\), what is the value of (n)?
B. (76)
(S_{n+2}-S_n=(n+1)+(n+2)=2n+3)। (2n+3=155) से (n=76)। / (S_{n+2}-S_n=(n+1)+(n+2)=2n+3). From (2n+3=155), (n=76).
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If \(S_n=2145\), what is the value of \(S_{n-3}\)?
A. (1953)
\(2145=S_{65}\), इसलिए \(S_{62}=\frac{62\cdot63}{2}=1953\)। (n) पहचानकर तीन कम करें। / \(2145=S_{65}\), so \(S_{62}=\frac{62\cdot63}{2}=1953\). Identify (n) and reduce it by three.
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