JEE Mains · Maths · STD 11 - 12. limits
If \(\lim \limits_{x \rightarrow 1} \frac{x+x^{2}+x^{3}+\ldots+x^{n}-n}{x-1}=820,(n \in N)\) then the value of \(n\) is equal to
- A \(35\)
- B \(45\)
- C \(40\)
- D \(50\)
Answer & Solution
Correct Answer
(C) \(40\)
Step-by-step Solution
Detailed explanation
\(\lim _{x \rightarrow 1} \frac{x+x^{2}+\ldots \ldots+x^{2}-n}{x-1}=820\) \(\Rightarrow \quad \lim _{x \rightarrow 1}\left(\frac{x-1}{x-1}+\frac{x^{2}-1}{x-1}+\ldots . . \frac{x^{n}-1}{x-1}\right)=820\) \(\Rightarrow 1+2+\ldots .+n=820\) \(\Rightarrow \quad n(n+1)=2 \times 820\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If \(^n{C_4},{\,^n}{C_5},\) and \({\,^n}{C_6},\) are in \(A.P.,\) then \(n\) can beJEE Mains 2019 Hard
- The locus of the mid-point of the line segment joining the focus of the parabola \(y^{2}=4 a x\) to a moving point of the parabola, is another parabola whose directrix isJEE Mains 2021 Hard
- The equation of the chord, of the ellipse \(\frac{x^2}{25}+\frac{y^2}{16}=1\), whose mid-point is \((3,1)\) is :JEE Mains 2025 Medium
- Let \(x=x(y)\) be the solution of the differential equation \(y^2 \mathrm{~d} x+\left(x-\frac{1}{y}\right) \mathrm{d} y=0\). If \(x(1)=1\), then \(x\left(\frac{1}{2}\right)\) is :JEE Mains 2025 Hard
- If \(\lim \limits_{x \rightarrow 0}\left\{\frac{1}{x^{8}}\left(1-\cos \frac{x^{2}}{2}-\cos \frac{x^{2}}{4}+\cos \frac{x^{2}}{2} \cos \frac{x^{2}}{4}\right)\right\}=2^{-k}\) then the value of \(k\) isJEE Mains 2020 Medium
- Let the point \(P\) of the focal chord \(P Q\) of the parabola \(y^2=16 x\) be \((1,-4)\). If the focus of the parabola divides the chord PQ in the ratio \(\mathrm{m}: \mathrm{n}\), \(\operatorname{gcd}(m, n)=1\), then \(m^2+n^2\) is equal to :JEE Mains 2025 Medium
More PYQs from JEE Mains
- A value of \(x\) for which \(\sin \,\left( {{{\cot }^{ - 1}}\,\left( {1 + x} \right)} \right) = \cos \,\left( {{{\tan }^{ - 1}}\,x} \right)\), isJEE Mains 2013 Medium
- The coefficient of \(x^{7}\) in the expression \((1+x)^{10}+x(1+x)^{9}+x^{2}(1+x)^{8}+\ldots+x^{10}\) isJEE Mains 2020 Hard
- The remainder when \((2021)^{2022}+(2022)^{ 2021 }\) is divided by \(7\) is.JEE Mains 2022 Hard
- If \(\cos \,\alpha + \cos \,\beta = \frac{3}{2}\) and \(\sin \,\alpha + \sin \,\beta = \frac{1}{2}\) and \(\theta \) is the the arithmetic mean of \(\alpha \) and \(\beta \) , then \(\sin \,2\theta + \cos \,2\theta \) is equal toJEE Mains 2015 Hard
- A vector \(\overrightarrow{ V }\) in the first octant is inclined to the \(x\) axis at \(60^{\circ}\), to the \(y\)-axis at \(45^{\circ}\) and to the z-axis at an acute angle. If a plane passing through the points \((\sqrt{2},-1,1)\) and \(( a , b , c )\), is normal to \(\overrightarrow{ v }\), thenJEE Mains 2023 Hard
- Let \(S\) be the set of all \(a \in N\) such that the area of the triangle formed by the tangent at the point \(P ( b , c ), b , c \in N\), on the parabola \(y ^2=2 ax\) and the lines \(x=b, y=0\) is \(16\) unit \(^2\), then \(\sum_{\text {aes }} a\) is equal to \(..........\).JEE Mains 2023 Hard