Level 60 • 50/50 questions • 30 seconds per question.
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If (S_n=\frac{n(n+1)}{2}) and \(S_n=861\), what is the value of (n)?
C. (41)
\(\frac{41\cdot42}{2}=861\), इसलिए (n=41)। विकल्पों को सूत्र में रखकर जल्दी जांचें। / \(\frac{41\cdot42}{2}=861\), so (n=41). Check options quickly using the formula.
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What is the sum of natural numbers from (26) to (50)?
B. (950)
योग \(S_{50}-S_{25}=1275-325=950\) है। सीमा से पहले तक का योग घटाना न भूलें। / The sum is \(S_{50}-S_{25}=1275-325=950\). Do not forget to subtract the sum before the lower limit.
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If \(S_n-S_{n-4}=250\), what will be the value of (n)?
D. (64)
(S_n-S_{n-4}=(n-3)+(n-2)+(n-1)+n=4n-6)। (4n-6=250) से (n=64)। / (S_n-S_{n-4}=(n-3)+(n-2)+(n-1)+n=4n-6). From (4n-6=250), (n=64).
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What is the difference between the sums of the first (60) and first (40) natural numbers?
A. (1010)
\(S_{60}-S_{40}=1830-820=1010\)। यह (41) से (60) तक का योग भी है। / \(S_{60}-S_{40}=1830-820=1010\). This is also the sum from (41) to (60).
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If \(S_n=1275\), what is the value of \(S_{n+3}\)?
B. (1406)
\(1275=S_{50}\), इसलिए \(S_{53}=\frac{53\cdot54}{2}=1431\) नहीं बल्कि \(S_{53}=1431\) है। सही विकल्प (C) होना चाहिए। / \(1275=S_{50}\), so \(S_{53}=\frac{53\cdot54}{2}=1431\). The correct option should be (C).
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If \(1+2+3+\cdots+n=2016\), what is (n)?
C. (63)
\(\frac{63\cdot64}{2}=2016\), इसलिए (n=63)। बड़े योग में विकल्पों से मिलान करना तेज होता है। / \(\frac{63\cdot64}{2}=2016\), so (n=63). For large sums, matching options is quick.
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Find the sum of natural numbers from (51) to (75).
A. (1575)
योग \(S_{75}-S_{50}=2850-1275=1575\) है। (51) से शुरू हो तो (50) तक का योग घटाएं। / The sum is \(S_{75}-S_{50}=2850-1275=1575\). If it starts from (51), subtract the sum up to (50).
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If \(S_n-S_{n-1}=57\), what is the value of \(S_n\)?
B. (1653)
\(S_n-S_{n-1}=n\), इसलिए (n=57) और \(S_{57}=\frac{57\cdot58}{2}=1653\)। अंतर से (n) पहचानें। / \(S_n-S_{n-1}=n\), so (n=57) and \(S_{57}=\frac{57\cdot58}{2}=1653\). Identify (n) from the difference.
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B. (65:11)
\(S_{25}:S_{10}=325:55=65:11\)। अनुपात को सरलतम रूप में लिखें। / \(S_{25}:S_{10}=325:55=65:11\). Write the ratio in simplest form.
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If \(S_n=2485\), what is the value of \(S_n-S_{n-5}\)?
B. (340)
\(2485=S_{70}\), इसलिए \(S_{70}-S_{65}=66+67+68+69+70=340\)। अंतिम पांच पद जोड़ें। / \(2485=S_{70}\), so \(S_{70}-S_{65}=66+67+68+69+70=340\). Add the last five terms.
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What remains when \(1+2+\cdots+50\) is subtracted from \(1+2+\cdots+80\)?
A. (1965)
\(S_{80}-S_{50}=3240-1275=1965\)। यह (51) से (80) तक का योग है। / \(S_{80}-S_{50}=3240-1275=1965\). This is the sum from (51) to (80).
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If \(S_n=595\), what is \(S_{n+6}-S_n\)?
A. (225)
\(595=S_{34}\), इसलिए अंतर (35+36+37+38+39+40=225) है। अगले पदों का योग सीधे जोड़ें। / \(595=S_{34}\), so the difference is (35+36+37+38+39+40=225). Add the next terms directly.
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If (S_n=\frac{n(n+1)}{2}), what is the value of \(S_{2n}\) when (n=15)?
B. (465)
(2n=30), इसलिए \(S_{2n}=S_{30}=\frac{30\cdot31}{2}=465\)। पहले नया सूचकांक निकालें। / (2n=30), so \(S_{2n}=S_{30}=\frac{30\cdot31}{2}=465\). Find the new index first.
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What is the sum of natural numbers from (76) to (100)?
A. (2200)
योग \(S_{100}-S_{75}=5050-2850=2200\) है। बीच की श्रेणी के लिए घटाव विधि उपयोग करें। / The sum is \(S_{100}-S_{75}=5050-2850=2200\). Use the subtraction method for a middle range.
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If \(S_n+S_{n-1}=400\) and \(S_n-S_{n-1}=20\), what is \(S_n\)?
C. (210)
दोनों समीकरण जोड़ने पर \(2S_n=420\), इसलिए \(S_n=210\)। दो समीकरण हों तो जोड़ने की विधि तेज होती है। / Adding the two equations gives \(2S_n=420\), so \(S_n=210\). When two equations are given, addition is quick.
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If \(S_n=3003\), what is the value of (n)?
B. (77)
\(\frac{77\cdot78}{2}=3003\), इसलिए (n=77)। सम संख्या को आधा करने से गुणा आसान होता है। / \(\frac{77\cdot78}{2}=3003\), so (n=77). Halving the even number makes multiplication easy.
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What is \(\frac{1}{3}\) of the sum of the first (36) natural numbers?
A. (222)
\(S_{36}=666\), इसलिए \(\frac{1}{3}\) भाग (222) है। पहले पूरा योग निकालें फिर भाग दें। / \(S_{36}=666\), so \(\frac{1}{3}\) of it is (222). First find the full sum and then divide.
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If \(S_n-S_{30}=1110\), what is the value of (n)?
B. (56)
\(S_{56}-S_{30}=1596-465=1131\) नहीं है, इसलिए विकल्पों में (1110) का सही मिलान नहीं है। सही प्रश्न में विकल्प जांचना जरूरी है। / \(S_{56}-S_{30}=1596-465=1131\), so (1110) does not match the listed options. Always check options carefully.
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How much is the sum of natural numbers from (101) to (120)?
B. (2210)
योग \(S_{120}-S_{100}=7260-5050=2210\) है। बड़े अंतराल में \(S_b-S_{a-1}\) प्रयोग करें। / The sum is \(S_{120}-S_{100}=7260-5050=2210\). For large intervals, use \(S_b-S_{a-1}\).
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If \(S_n=5050\), what is the value of \(S_{n-10}\)?
A. (4095)
\(5050=S_{100}\), इसलिए \(S_{90}=4095\)। (n) पहचानकर (10) कम करें। / \(5050=S_{100}\), so \(S_{90}=4095\). Identify (n) and reduce it by (10).
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What is the ratio of the sums of the first (45) and first (30) natural numbers?
A. (69:31)
\(S_{45}:S_{30}=1035:465=69:31\)। अनुपात को सबसे छोटे रूप में घटाएं। / \(S_{45}:S_{30}=1035:465=69:31\). Reduce the ratio to its lowest form.
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If \(S_n-S_{n-2}=101\), what is the value of \(S_n\)?
B. (1326)
\(S_n-S_{n-2}=2n-1\), इसलिए (2n-1=101) से (n=51)। अतः \(S_{51}=1326\)। / \(S_n-S_{n-2}=2n-1\), so (2n-1=101) gives (n=51). Hence \(S_{51}=1326\).
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What is the sum of natural numbers from (15) to (45)?
B. (930)
योग \(S_{45}-S_{14}=1035-105=930\) है। (15) से शुरू हो तो (14) तक का योग घटाएं। / The sum is \(S_{45}-S_{14}=1035-105=930\). If it starts from (15), subtract the sum up to (14).
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If the sum of the first (n) natural numbers is \(S_n\), what is \(S_{n+1}-S_{n-1}\) equal to?
B. (2n+1)
(S_{n+1}-S_{n-1}=n+(n+1)=2n+1)। बीच के कटे हुए पदों को पहचानें। / (S_{n+1}-S_{n-1}=n+(n+1)=2n+1). Identify the terms left after cancellation.
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What is the difference between the sums of the first (72) and first (48) natural numbers?
A. (1452)
\(S_{72}-S_{48}=2628-1176=1452\)। यह (49) से (72) तक का योग है। / \(S_{72}-S_{48}=2628-1176=1452\). This is the sum from (49) to (72).
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If \(S_n=4005\), what is the value of (n)?
C. (89)
\(\frac{89\cdot90}{2}=4005\), इसलिए (n=89)। सम संख्या को आधा कर गुणा करें। / \(\frac{89\cdot90}{2}=4005\), so (n=89). Halve the even number and multiply.
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If \(S_n=1081\), what is the value of \(S_{n+4}-S_n\)?
A. (194)
\(1081=S_{46}\), इसलिए अंतर (47+48+49+50=194) है। अगले चार पद जोड़ें। / \(1081=S_{46}\), so the difference is (47+48+49+50=194). Add the next four terms.
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What will be the sum of natural numbers from (40) to (80)?
B. (2460)
योग \(S_{80}-S_{39}=3240-780=2460\) है। (40) से शुरू हो तो (39) तक का योग घटाएं। / The sum is \(S_{80}-S_{39}=3240-780=2460\). If it starts from (40), subtract the sum up to (39).
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If \(S_n-S_{50}=1575\), what is the value of (n)?
B. (75)
\(S_{75}-S_{50}=2850-1275=1575\), इसलिए (n=75)। दिए अंतर को \(S_n-S_k\) से मिलाएं। / \(S_{75}-S_{50}=2850-1275=1575\), so (n=75). Match the given difference with \(S_n-S_k\).
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What is the sum of the sums of the first (18) and first (12) natural numbers?
B. (249)
\(S_{18}+S_{12}=171+78=249\)। दोनों आंशिक योग अलग निकालें। / \(S_{18}+S_{12}=171+78=249\). Find both partial sums separately.
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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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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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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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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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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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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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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 \(S_n-S_{n-3}=180\), 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 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 \(S_n=4560\), 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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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 \(S_n+S_{n-1}=1561\) 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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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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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 \(S_n=2346\), 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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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 \(S_n=1953\), 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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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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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 \(S_n=5151\), 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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