SATHEE: Sequences and Series 2 Question 11

Sequences and Series 2 Question 11

Passage Based Problems

Read the following passage and answer the questions.

Passage

Let $V_{r}$ denotes the sum of the first $r$ terms of an arithmetic progression (AP) whose first term is $r$ and the common difference is $(2 r - 1)$ . Let $T_{r} = V_{r + 1} - V_{r}$ and $Q_{r} = T_{r + 1} - T_{r}$ for $r = 1, 2, \dots$

(2007, 8M)

12.

The sum $V_{1} + V_{2} + \dots + V_{n}$ is

(a) $\frac{1}{12} n (n + 1) (3 n^{2} - n + 1)$

(b) $\frac{1}{12} n (n + 1) (3 n^{2} + n + 2)$

(d) $\frac{1}{3} (2 n^{3} - 2 n + 3)$

Show Answer

Answer:

Correct Answer: 12. (b)

Solution:

Here, $V_{r} = \frac{r}{2} [2 r + (r - 1) (2 r - 1)] = \frac{1}{2} (2 r^{3} - r^{2} + r)$

$∴ Σ V_{r} = \frac{1}{2} [2 Σ r^{3} - Σ r^{2} + Σ r]$

$= \frac{1}{2} 2 [{\frac{n (n + 1)}{2}}^{2}] - \frac{n (n + 1) (2 n + 1)}{6} + \frac{n (n + 1)}{2}$

$\Rightarrow \frac{n (n + 1)}{12} [3 n (n + 1) - (2 n + 1) + 3]$

$= \frac{1}{12} n (n + 1) (3 n^{2} + n + 2)$

Sequences and Series 2 Question 11

12.

Answer:

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Sequences and Series 2 Question 11

12.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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