Let s1, s2, s3,…..,s10 respectively be the sum to 12 terms of 10 A.P.s whose first terms are 1,2,3, …. ,10 and the common differences are 1, 3, 5, …. , 19 respectively. Then \(10∑^{10}_{i=1} s_i\) is
- 7260
- 7220
- 7360
- 7380
The Correct Option is A
Solution and Explanation
Denote by \(s_k\) the sum of 12 terms of the \(k\)-th arithmetic progression, whose first term is \(k\) and common difference is \(2k-1\). We recall that the sum of 12 terms of an A.P. with first term \(a\) and common difference \(d\) is \[ S_{12} = \frac{12}{2} [2a + (12 - 1)d] = 6(2a + 11d). \] Hence for the \(k\)-th progression, \[ s_k = 6(2k + 11(2k - 1)) = 6(2k + 22k - 11) = 6(24k - 11) = 144k - 66. \] We wish to find \[ \sum_{k=1}^{10} s_k = \sum_{k=1}^{10} (144k - 66) = 144 \sum_{k=1}^{10} k - 66 \sum_{k=1}^{10} 1. \] Recall \[ \sum_{k=1}^{10} k = \frac{10 \cdot 11}{2} = 55, \] and \[ \sum_{k=1}^{10} 1 = 10. \] Thus \[ \sum_{k=1}^{10} s_k = 144 \cdot 55 - 66 \cdot 10 = 7920 - 660 = 7260. \] Hence, \[ \boxed{ \sum_{k=1}^{10} s_k = 7260. } \]
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main types of differential equations Questions
- Let the system of equations \(x+2y+3z = 5\), \(2x+3y+z = 9\), \(4x+3y+λz = μ\) have an infinite number of solutions. Then \(λ + 2μ\) is equal to
- Let \(y=y(t)\) be a solution of the differential equation\(\frac{d y}{d t}+\alpha y=\gamma e^{-\beta t}\) where, \(\alpha > 0, \beta>0\) and \(\gamma > 0\). Then \(\displaystyle\lim _{t \rightarrow \infty} y(t)\)
The area enclosed by the closed curve $C$ given by the differential equation $\frac{d y}{d x}+\frac{x+a}{y-2}=0, y(1)=0$ is $4 \pi$.
Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$-axis If normals at $P$ and $Q$ on the curve $C$ intersect $x$-axis at points $R$ and $S$ respectively, then the length of the line segment $R S$ is
The statement
\((p⇒q)∨(p⇒r) \)
is NOT equivalent toLet α, β(α > β) be the roots of the quadratic equation x2 – x – 4 = 0.
If \(P_n=α^n–β^n, n∈N\) then \(\frac{P_{15}P_{16}–P_{14}P_{16}–P_{15}^2+P_{14}P_{15}}{P_{13}P_{14}}\)
is equal to _______.
Top JEE Main Questions
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,