KCET · Maths · Sequences and Series
\(\frac{1}{2 \cdot 5}+\frac{1}{5 \cdot 8}+\frac{1}{8 \cdot 11}+\ldots \frac{1}{(3 n-1)(3 n+2)}\) is equal to
- A \(\frac{n}{6 n-4}\)
- B \(\frac{n}{6 n+3}\)
- C \(\frac{n}{6 n+4}\)
- D \(\frac{n+1}{6 n+4}\)
Answer & Solution
Correct Answer
(C) \(\frac{n}{6 n+4}\)
Step-by-step Solution
Detailed explanation
Given, \(\frac{1}{2 \cdot 5}+\frac{1}{5 \cdot 8}+\frac{1}{8 \cdot 11}+\ldots\)
\[
\begin{array}{r}
5 \cdot 8 \quad 8 \cdot 11 \\
=\frac{1}{3}\left[\left(\frac{1}{2}-\frac{1}{5}\right)+\frac{1}{(3 n-1)(3 n+2)}\right. \\
+\ldots+\left(\frac{1}{5}-\frac{1}{8}\right)+\left(\frac{1}{8}-\frac{1}{11}\right) \\
\left.\left.+\ldots-\frac{1}{3 n+2}\right)\right]
\end{array}
\]
\[
\begin{aligned}
&=\frac{1}{3}\left[\frac{1}{2}-\frac{1}{3 n+2}\right] \\
&=\frac{3 n+2-2}{6(3 n+2)}=\frac{n}{6 n+4}
\end{aligned}
\]
\[
\begin{array}{r}
5 \cdot 8 \quad 8 \cdot 11 \\
=\frac{1}{3}\left[\left(\frac{1}{2}-\frac{1}{5}\right)+\frac{1}{(3 n-1)(3 n+2)}\right. \\
+\ldots+\left(\frac{1}{5}-\frac{1}{8}\right)+\left(\frac{1}{8}-\frac{1}{11}\right) \\
\left.\left.+\ldots-\frac{1}{3 n+2}\right)\right]
\end{array}
\]
\[
\begin{aligned}
&=\frac{1}{3}\left[\frac{1}{2}-\frac{1}{3 n+2}\right] \\
&=\frac{3 n+2-2}{6(3 n+2)}=\frac{n}{6 n+4}
\end{aligned}
\]
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
- The angles of a triangle are in A.P and the greatest angle is double the least angle, then sine of the third angle isKCET 2026 Easy
- Integrating factor of \( x \frac{d y}{d x}-y=x^{4}-3 x \) isKCET 2016 Easy
- If \(y=\sin x \cdot \sin 2 x \cdot \sin 3 x \ldots \sin n x\), then \(y^{\prime}\) isKCET 2011 Medium
- If \( |\bar{a}| \) and \( |\bar{b}| \) are mutually perpendicular unit vectors, then
\( (3|\bar{a}|+2|\bar{b}|) \cdot(5|\bar{a}|-6|\bar{b}|)= \)KCET 2018 Medium - If the side of a cube is increased by \(5 \%\), then the surface area of a cube is increased byKCET 2020 Easy
- If \( \mathrm{A} \) is a matrix of order \( \mathrm{m} \times \mathrm{n} \) and \( \mathrm{B} \) is a matrix such that \( \mathrm{AB}^{\prime} \) and \( \mathrm{B}^{\prime} \mathrm{A} \) are both defined, the
order of the matrix \( B \) isKCET 2016 Medium
More PYQs from KCET
- Match the hormone with its site of production:
Hormone Site of production A. hCG and hPL (i) Ovary B. Progesterone (ii) Placenta C. Androgens (iii) Corpus luteum D. Relaxin (iv) Leydig cells KCET 2025 Hard - In a nuclear reactor heavy nuclei is not used as moderators becauseKCET 2021 Hard
- In the given diagram identify the parts labelled as A, B, C and D.
KCET 2021 Hard - The kinetic energy of an electron gets tripled, then the de-Broglie wavelength associated with it changes by a factorKCET 2009 Medium
- The diagram of the section of a maize grain is given below. Identify the parts labelled A, B, C and D.
KCET 2009 Easy - Specific conductance of \(0.1 \mathrm{M} \mathrm{HNO}_3\) is \(63 \times 10^{-2} \mathrm{ohm}^{-1} \mathrm{~cm}^{-1}\). The molar conductance of the solution isKCET 2022 Medium