JEE Advanced · Mathematics · 5. Sequences & Series
Suppose four distinct positive numbers \(a_1, a_2, a_3\) and \(a_4\) are in GP. Let \(b_1=a_1\), \(b_2=b_1+a_2, b_3=b_2+a_3\) and \(b_4=b_3+a_4\).
Statement 1 The numbers \(b_1, b_2, b_3\) and \(b_4\) are neither in AP nor in GP.
Statement 2 The numbers \(b_1, b_2, b_3\) and \(b_4\) are in HP.
- A Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
- B Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
- C Statement 1 is true, Statement 2 is false.
- D Statement 1 is false, Statement 2 is true
Answer & Solution
Correct Answer
(C) Statement 1 is true, Statement 2 is false.
Step-by-step Solution
Detailed explanation
Let \(a_1=1, a_2=2, a_3=4, a_4=8\)
\(\therefore b_1=1, b_2=3, b_3=7, b_4=15\)
Clearly, \(b_1, b_2, b_3\) and \(b_4\) are not in HP.
Statement 2 is false.
\(\therefore b_1=1, b_2=3, b_3=7, b_4=15\)
Clearly, \(b_1, b_2, b_3\) and \(b_4\) are not in HP.
Statement 2 is false.
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 Mathematics
- The value of
\(6+\log _{\frac{3}{2}}\)\(\left(\frac{1}{3 \sqrt{2}} \sqrt{4-\frac{1}{3 \sqrt{2}} \sqrt{4-\frac{1}{3 \sqrt{2}} \sqrt{4-\frac{1}{3 \sqrt{2}}} \cdots}}\right)\) isJEE Advanced 2012 Medium - Paragraph:
Consider the circle \(x^2+y^2=9\) and the parabola \(y^2=8 x\). They intersect at \(P\) and \(Q\) in the first and the fourth quadrants, respectively. Tangents to the circle at \(P\) and \(Q\) intersect the \(\mathrm{X}\)-axis at \(R\) and tangents to the parabola at \(P\) and \(Q\) intersect the \(\mathrm{X}\)-axis at \(S\).Question:
The radius of the incircle of the \(\triangle P Q R\) isJEE Advanced 2007 Easy - Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and more over the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done isJEE Advanced 2014 Medium
- Let be the cube with the set of vertices . Let be the set of all twelve lines containing the diagonals of the six faces of the cube . Let be the set of all four lines containing the main diagonals of the cube ; for instance, the line passing through the vertices and is in . For lines and , let denote the shortest distance between them. Then the maximum value of , as varies over and varies over , isJEE Advanced 2023 Easy
- Perpendiculars are drawn from points on the line to the plane . The feet of perpendiculars lie on the lineJEE Advanced 2013 Easy
- Consider the parabola \(y^2=8 x\). Let \(\Delta_1\) be the area of the triangle formed by the end points of its latusrectum and the point \(P\left(\frac{1}{2}, 2\right)\) on the parabola and \(\Delta_2\) be the area of the triangle formed by drawing tangents at \(P\) and at the end points of the latusrectum. Then, \(\frac{\Delta_1}{\Delta_2}\) isJEE Advanced 2011 Hard
More PYQs from JEE Advanced
- Let \(P, Q, R\) and \(S\) be the points on the plane with position vectors \(-2 \hat{\mathbf{i}}-\hat{\mathbf{j}}, 4 \hat{\mathbf{i}}, 3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\) an \(\mathrm{d}-3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}\) respectively. The quadrilateral \(P Q R S\) must be aJEE Advanced 2010 Easy
- Paragraph:
Consider the polynomial \(f(x)=1+2 x+3 x^2+4 x^3\). Let \(s\) be the sum of all distinct real roots of \(f(x)\) and let \(t=|s|\).Question:
The area bounded by the curve \(y=f(x)\) and the lines \(x=0, y=0\) and \(x=t\), lies in the intervalJEE Advanced 2010 Hard - Let thenJEE Advanced 2017 Easy
- Paragraph:
\(P\) and \(Q\) are isomers of dicarboxylic acid \(\mathrm{C}_{4} \mathrm{H}_{4} \mathrm{O}_{4}\). Both decolorize \(\mathrm{Br}_{2} / \mathrm{H}_{2} \mathrm{O}\). On heating, \(\boldsymbol{P}\) forms the cyclic anhydride.
Upon treatment with dilute alkaline \(\mathrm{KMnO}_{4}, \boldsymbol{P}\) as well as \(\boldsymbol{Q}\) could produce one or more than one from \(\boldsymbol{S}, \boldsymbol{T}\) and \(\boldsymbol{U}\).
Question:
Compounds formed from \(\boldsymbol{P}\) and \(\boldsymbol{Q}\) are, respectivelyJEE Advanced 2013 Medium - The total number of possible isomers for is___.JEE Advanced 2021 Medium
- A tetrapeptide has -COOH group on alanine. This produces glycine (Gly), Valine (Val), Phenyl alanine (Phe) and Alanine (Ala) on complete hydrolysis. For this tetrapeptide, the number of possible sequences (primary structures) with -NH2 group attached to a chiral center isJEE Advanced 2013 Hard