JEE Mains · Maths · STD 11 - 8. sequence and series
\( \text { If } S(x)=(1+x)+2(1+x)^2+3(1+x)^3+\ldots . \) \( +60(1+x)^{60}, x \neq 0 \text {, and }(60)^2 S(60)=a(b)^b+b\) where \(a, b N\), then \((a+b)\) equal to ...............
- A \(3214\)
- B \(1495\)
- C \(3660\)
- D \(3654\)
Answer & Solution
Correct Answer
(C) \(3660\)
Step-by-step Solution
Detailed explanation
\( S(x)=(1+x)+2(1+x)^2+3(1+x)^3+. .+60(1+x)^{60} \) \((1+x) S=(1+x)^2+\ldots \ldots . . \quad 59(1+x)^{60}+60(1+x)^{61} \) \(-x S=\frac{(1+x)(1+x)^{60}-1}{x}-60(1+x)^{61}\) Put \(\mathrm{x}=60\) \(-60 \mathrm{~S}=\frac{61\left((61)^{60}-1\right)}{60}-60(61)^{61}\) on solving…
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
- Consider the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x)=\frac{2 x}{\sqrt{1+9 x^2}}\). If the composition of \(f, \underbrace{(f \circ f \circ f \circ \ldots \circ f)}_{10 \text { times }}(x)=\frac{2^{10} x}{\sqrt{1+9 \alpha x^2}}\), then the value of \(\sqrt{3 \alpha+1}\) is equal to ...........JEE Mains 2024 Hard
- If \(\frac{d y}{d x}=\frac{2^{x} y+2^{y} \cdot 2^{x}}{2^{x}+2^{x+y} \log _{e} 2}, y(0)=0\), then for \(y=1\) the value of \(x\) lies in the interval:JEE Mains 2021 Hard
- In a group of \(400\) people, \(160\) are smokers and nonvegetarian; \(100\) are smokers and vegetarian and the remaining \(140\) are non-smokers and vegetarian. Their chances of getting a particular chest disorder are \(35\, \%, 20 \,\%\) and \(10 \,\%\) respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is ...... .JEE Mains 2021 Medium
- Let the sets \(A\) and \(B\) denote the domain and range respectively of the function \(f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}\) where \(\lceil x \rceil\) denotes the smallest integer greater than or equal to \(x\). Then among the statements \(( S 1): A \cap B =(1, \infty)-N\) and \(( S 2): A \cup B=(1, \infty)\)JEE Mains 2023 Hard
- The value of \(\sum_{n=1}^{100} \int_{n-1}^{n} e^{x-[x]} d x,\) where \([x]\) is the greatest integer \(\leq x ,\) isJEE Mains 2021 Medium
- For three unit vectors \(\vec{a}, \vec{b}, \vec{c}\) satisfying \({|\vec{a}-\vec{b}|^{2}}+{|\vec{b}-\vec{c}|^{2}}+{|\vec{c}-\vec{a}|^{2}}=9\) and \({|2\vec{a}+k\vec{b}+k\vec{c}|}=3\), the positive value of k is:JEE Mains 2026 Medium
More PYQs from JEE Mains
- If equation of the plane that contains the point \((-2,3,5)\) and is perpendicular to each of the planes \(2 x+4 y+5 z=8\) and \(3 x-2 y+3 z=5\) is \(\alpha x+\beta y+\gamma z+97=0\) then \(\alpha+\beta+\gamma=...........\).JEE Mains 2023 Hard
- Let \(\mathrm{E}: \frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}=1, \mathrm{a}\gt\mathrm{b}\) and \(\mathrm{H}: \frac{x^2}{\mathrm{~A}^2}-\frac{y^2}{\mathrm{~B}^2}=1\). Let the distance between the foci of E and the foci of \(H\) be \(2 \sqrt{3}\). If \(a-A=2\), and the ratio of the eccentricities of \(E\) and \(H\) is \(\frac{1}{3}\), then the sum of the lengths of their latus rectums is equal to:JEE Mains 2025 Hard
- Let \(\alpha, \beta, \gamma\) be the three roots of the equation \(x ^3+ bx + c =0\). If \(\beta \gamma=1=-\alpha\), then \(b^3+2 c^3-3 \alpha^3-6 \beta^3-8 \gamma^3\) is equal to \(......\).JEE Mains 2023 Hard
- A square is inscribed in the circle \(x^2+y^2-10 x-6 y+30=0\). One side of this square is parallel to \(y=x+3\). If \(\left(x_i, y_i\right)\) are the vertices of the square, then \(\sum\left(\mathrm{x}_{\mathrm{i}}^2+\mathrm{y}_{\mathrm{i}}^2\right)\) is equal to :JEE Mains 2024 Hard
- If \(S\) is the set of distinct values of \('b'\) for which the following system of linear equations \(x + y + z = 1;x + ay + z = 1;ax + by + z = 0\) has no solution , then \(S\) is :JEE Mains 2017 Hard
- If \(\theta \in\left[-\frac{7 \pi}{6}, \frac{4 \pi}{3}\right]\), then the number of solutions of \(\sqrt{3} \operatorname{cosec}^2 \theta-2(\sqrt{3}-1) \operatorname{cosec} \theta-4=0\), is equal toJEE Mains 2025 Medium