Question:
The value of \( a = 3 \), the sum of first 4 terms is equal to the one fifth of the sum of next 4 terms in A.P., then find the sum of the first 20 terms.
The value of \( a = 3 \), the sum of first 4 terms is equal to the one fifth of the sum of next 4 terms in A.P., then find the sum of the first 20 terms.
Show Hint
In A.P. problems, always use the formula for the sum of terms and apply the given conditions to solve for the unknown common difference or terms.
Updated On: Jan 14, 2026
- 340
- 360
- 380
- 400
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Let the first term of the arithmetic progression (A.P.) be \( a = 3 \), and the common difference be \( d \). The sum of the first 4 terms is:
\[
S_4 = 4a + 6d = 4(3) + 6d = 12 + 6d.
\]
The sum of the next 4 terms (terms 5 to 8) is:
\[
S_8 - S_4 = (8a + 28d) - (4a + 6d) = 4a + 22d = 12 + 22d.
\]
It is given that:
\[
S_4 = \frac{1}{5}(S_8 - S_4).
\]
Substitute the expressions for \( S_4 \) and \( S_8 - S_4 \):
\[
12 + 6d = \frac{1}{5}(12 + 22d).
\]
Solve for \( d \):
\[
5(12 + 6d) = 12 + 22d \quad \Rightarrow \quad 60 + 30d = 12 + 22d \quad \Rightarrow \quad 8d = -48 \quad \Rightarrow \quad d = -6.
\]
Now, find the sum of the first 20 terms:
\[
S_{20} = \frac{20}{2} \left[ 2a + (n-1)d \right] = 10 \left[ 2(3) + 19(-6) \right] = 10 \left[ 6 - 114 \right] = 10 \times (-108) = 360.
\]
Thus, the correct answer is option (2) 360.
Was this answer helpful?
0
0
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_____
View More Questions
Top JEE Main Miscellaneous Questions
- Let \( D = \{a, b, c\} \). How many distinct ways can \( D \) be partitioned into non-empty subsets, representing equivalence relations?
- A bag contains 4 red balls, 3 blue balls, and 2 green balls. Two balls are picked at random. What is the probability that both balls are of the same color?
- A circle passes through the points \( (1, 1) \) and \( (2, -1) \), and its center lies on the line \( x + y = 4 \). What is the radius of the circle?
- If |A| = 2, B = adj(adj(2A)), A is a 3×3 matrix, and tr(A) = 3, what is tr(B) + |B|?
- From the English alphabets, 5 letters are chosen and arranged in alphabetical order. The total number of ways in which the middle letter is M is:
View More Questions
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- 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 ______.
View More Questions