JEE Mains · Maths · STD 11 - 8. sequence and series
The sum of the first \(20\) terms of the series \(1 + \frac{3}{2} + \frac{7}{4} + \frac{{15}}{8} + \frac{{31}}{{16}} + ...\) is?
- A \(38 + \frac{1}{{{2^{20}}}}\)
- B \(39 + \frac{1}{{{2^{19}}}}\)
- C \(39 + \frac{1}{{{2^{20}}}}\)
- D \(38 + \frac{1}{{{2^{19}}}}\)
Answer & Solution
Correct Answer
(D) \(38 + \frac{1}{{{2^{19}}}}\)
Step-by-step Solution
Detailed explanation
The general term of the given series \( = \frac{{2 \times {2^r} - 1}}{{{2^r}}}\), Where \(r \ge 0\) \(\therefore \) req.sum \( = 1 + \sum\limits_{r = 1}^{19} {\frac{{2 \times {2^r} - 1}}{{{2^r}}}} \) Now,…
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 two tangents drawn from a point \((\alpha, \beta)\) lying on the ellipse \(25 x^{2}+4 y^{2}=1\) to the parabola \(y^{2}=4 x\) are such that the slope of one tangent is four times the other, then the value of \((10 \alpha+5)^{2}+\left(16 \beta^{2}+50\right)^{2}\) equalsJEE Mains 2022 Hard
- If \(\frac{1}{2 \cdot 3^{10}}+\frac{1}{2^{2} \cdot 3^{9}}+\ldots \frac{1}{2^{10} \cdot 3}=\frac{K}{2^{10} \cdot 3^{10}}\), then the remainder when \(K\) is divided by \(6\) isJEE Mains 2022 Hard
- The value of the integral \(\int \frac{\sin \theta \cdot \sin 2 \theta\left(\sin ^{6} \theta+\sin ^{4} \theta+\sin ^{2} \theta\right) \sqrt{2 \sin ^{4} \theta+3 \sin ^{2} \theta+6}}{1-\cos 2 \theta} d \theta\) is (where \(c\) is a constant of integration)JEE Mains 2021 Hard
- Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined as \(f(x) = \dfrac{2x^2 - 3x + 2}{3x^2 + x + 3}\). Then \(f\) is :JEE Mains 2026 Medium
- If \(m\) is the slope of a common tangent to the curves \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) and \(x^{2}+y^{2}=12\), then \(12\; m ^{2}\) is equal toJEE Mains 2022 Hard
- If \(\frac{3+i \sin \theta}{4-i \cos \theta}, \theta \in[0,2 \pi],\) is a real number, then an argument of \(\sin \theta+\mathrm{i} \cos \theta\) isJEE Mains 2020 Hard
More PYQs from JEE Mains
- Let \(\mathrm{M}\) denote the median of the following frequency distribution then \(20\) \(M\) is equal to :
Class \(0-4\) \(4-8\) \(8-12\) \(12-16\) \(16-20\) Freq \(3\) \(9\) \(10\) \(8\) \(6\) JEE Mains 2024 Hard - Let \(5 f(x)+4 f\left(\frac{1}{x}\right)=\frac{1}{x}+3, x > 0\). Then \(18 \int \limits_1^2 f(x) d x\) is equal to:JEE Mains 2023 Hard
- Let \(f : R \rightarrow R\) be a differentiable function such that \(f \left(\frac{\pi}{4}\right)=\sqrt{2}, f \left(\frac{\pi}{2}\right)=0\) and \(f ^{\prime}\left(\frac{\pi}{2}\right)=1\) and let \(g(x)=\int\limits_{x}^{\pi / 4}\left(f^{\prime}(t) \sec t+\tan t \operatorname{sectf}(t)\right) d t\) for \(x \in\left[\frac{\pi}{4}, \frac{\pi}{2}\right)\). Then \(\lim\limits _{ x \rightarrow\left(\frac{\pi}{2}\right)^{-}} g ( x )\) is equal toJEE Mains 2022 Hard
- Let \(\{x\}\) and \([x]\) denote the fractional part of \(x\) and the greatest integer \(\leq x\) respectively of a real number \(x\). If \(\int \limits_{0}^{n}\{x\} d x, \int \limits_{0}^{n}[x] d x\) and \(10\left( n ^{2}- n \right),( n \in N , n >1)\) are three consecutive terms of a \(G.P.\), then \(n\) is equal toJEE Mains 2020 Hard
- The sum of the rational terms in the binomial expansion of \({\left( {{2^{\frac{1}{2}}} + {3^{\frac{1}{5}}}} \right)^{10}}\) isJEE Mains 2013 Hard
- \(\operatorname{Lim}_{x \rightarrow 0} \frac{e-(1+2 x)^{\frac{1}{2 x}}}{x}\) is equal to :JEE Mains 2024 Hard