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.
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 \(S_n-S_{50}=1575\), 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 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 \(S_n=1081\), 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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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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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 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 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 \(S_n-S_{n-2}=101\), 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 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 \(S_n=5050\), 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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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 \(S_n-S_{30}=1110\), 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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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 \(S_n=3003\), 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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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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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=\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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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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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 \(S_n=2485\), 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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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 \(S_n-S_{n-1}=57\), 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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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 \(1+2+3+\cdots+n=2016\), 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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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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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 \(S_n-S_{n-4}=250\), 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 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=\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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