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If the sum of first p terms of an A.P. is equal to the sum of the firs...

If the sum of first p terms of an A.P. i | Class 11 Mathematics Chapter Sequence and Series, Sequence and Series NCERT Solutions

Welcome to the NCERT Solutions for Class 11 Mathematics - Chapter Sequence and Series. This page offers a step-by-step solution to the specific question from Exercise 2, Question 10: . With detailed answers and explanations for each chapter, students can strengthen their understanding and prepare confidently for exams. Ideal for CBSE and other board students, this resource will simplify your study experience.

Question 10:

If the sum of first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first (p + q) terms.

Answer:

Let a and d be the first term and the common difference of the A.P. respectively.

Here,

\begin{align} S_p = \frac{p}{2}\left[2a+(p-1)d\right]\end{align}

\begin{align} S_q = \frac{q}{2}\left[2a+(q-1)d\right]\end{align}

According to the given condition,

\begin{align} \frac{p}{2}\left[2a+(p-1)d\right]=\frac{q}{2}\left[2a+(q-1)d\right]\end{align}

\begin{align} ⇒p\left[2a+(p-1)d\right]=q\left[2a+(q-1)d\right]\end{align}

\begin{align} ⇒2ap + pd(p-1)=2aq+qd(q-1)\end{align}

\begin{align} ⇒2a(p-q) +d[p(p-1)-q(q-1)]=0\end{align}

\begin{align} ⇒2a(p-q) +d[p^2 -p -q^2 +q]=0\end{align}

\begin{align} ⇒2a(p-q) +d[(p-q)(p+q)-(p-q)]=0\end{align}

\begin{align} ⇒2a(p-q) +d[(p-q)(p+q-1)]=0\end{align}

\begin{align} ⇒2a +d(p+q-1)=0\end{align}

\begin{align} ⇒d=\frac{-2a}{p+q-1} \;\;\;\;...(1)\end{align}

\begin{align} \therefore S_{p+q}=\frac{p+q}{2}\left[2a+(p+q-1).d\right] \end{align}

\begin{align} ⇒ S_{p+q}=\frac{p+q}{2}\left[2a+(p+q-1).\left(\frac{-2a}{p+q-1}\right)\right] \;\;\;\; [From (1)] \end{align}

\begin{align}=\frac{p+q}{2}\left[2a-2a\right]\end{align}

\begin{align}=0\end{align}

Thus, the sum of the first (p + q) terms of the A.P. is 0.

Other Questions of Chapter Sequence and Series

How many terms of G.P. 3, 3², 3³, … are needed to give the sum 120?

Find the sum to n terms of the series whose nth terms is given by (2n – 1)²

The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.

If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.

Find a G.P. for which sum of the first two terms is –4 and the fifth term is 4 times the third term.

Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.

Let the sum of n, 2n, 3n terms of an A.P. be S₁, S₂ and S₃, respectively, show that S₃ = 3 (S₂– S₁)

If the p^th, q^th and r^th terms of a G.P. are a, b and c, respectively. Prove that a^q-rb^r-pc^p-q=1

Find the sum to 20 terms in the geometric progression 0.15, 0.015, 0.0015 …

Q: Write the first five terms of the sequences whose nth term is a_n=n(n+2)

Popular questions of class 11 Mathematics

A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

Q: Write the following sets in roster form:
(i) A = {x: x is an integer and - 3 < x < 7}.
(ii) B = {x: x is a natural number less than 6}.
(iii) C = {x: x is a two-digit natural number such that the sum of its digits is 8}
(iv) D = {x: x is a prime number which is divisor of 60}.
(v) E = The set of all letters in the word TRIGONOMETRY.
(vi) F = The set of all letters in the word BETTER.

Q: Write the following sets in the set-builder form:
(i) (3, 6, 9, 12)
(ii) {2, 4, 8, 16, 32}
(iii) {5, 25, 125, 625}
(iv) {2, 4, 6 upto infinity}
(v) {1, 4, 9, upto 100}

How many 4-letter code can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?

Name the octants in which the following points lie:

(1, 2, 3), (4, –2, 3), (4, –2, –5), (4, 2, –5), (–4, 2, –5), (–4, 2, 5), (–3, –1, 6), (2, –4, –7)

A point is in the XZ-plane. What can you say about its y-coordinate?

How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that

(i) repetition of the digits is allowed?

(ii) repetition of the digits is not allowed?

Which of the following sentences are statements? Give reasons for your answer.

(i) There are 35 days in a month.

(ii) Mathematics is difficult.

(iii) The sum of 5 and 7 is greater than 10.

(iv) The square of a number is an even number.

(v) The sides of a quadrilateral have equal length.

(vi) Answer this question.

(vii) The product of (–1) and 8 is 8.

(viii) The sum of all interior angles of a triangle is 180°.

(ix) Today is a windy day.

(x) All real numbers are complex numbers.

An experiment consists of tossing a coin and then throwing it second time if a head occurs. If a tail occurs on the first toss, then a die is rolled once. Find the sample space.

A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are there?

Recently Viewed Questions of Class 11 Mathematics

The base of an equilateral triangle with side 2a lies along the y-axis such that the mid-point of the base is at the origin. Find vertices of the triangle.

A point is in the XZ-plane. What can you say about its y-coordinate?

A point is on the x-axis. What are its y-coordinates and z-coordinates?

Find the equation of the circle with centre (0, 2) and radius 2

Solve 24x < 100, when
(i) x is a natural number. (ii) x is an integer.

Prove the following by using the principle of mathematical induction for all n ∈ N:

n (n + 1) (n + 5) is a multiple of 3.

Which of the following sentences are statements? Give reasons for your answer.

(i) There are 35 days in a month.

(ii) Mathematics is difficult.

(iii) The sum of 5 and 7 is greater than 10.

(iv) The square of a number is an even number.

(v) The sides of a quadrilateral have equal length.

(vi) Answer this question.

(vii) The product of (–1) and 8 is 8.

(viii) The sum of all interior angles of a triangle is 180°.

(ix) Today is a windy day.

(x) All real numbers are complex numbers.

An experiment consists of rolling a die and then tossing a coin once if the number on the die is even. If the number on the die is odd, the coin is tossed twice. Write the sample space for this experiment.

A coin is tossed. If it shows a tail, we draw a ball from a box which contains 2 red and 3 black balls. If it shows head, we throw a die. Find the sample space for this experiment.

Find the derivative of x² – 2 at x = 10.

Study Tips for Answering NCERT Questions:

NCERT questions are designed to test your understanding of the concepts and theories discussed in the chapter. Here are some tips to help you answer NCERT questions effectively:

Read the question carefully and focus on the core concept being asked.
Reference examples and data from the chapter when answering questions about Sequence and Series.
Review previous year question papers to get an idea of how such questions may be framed in exams.
Practice answering questions within the time limit to improve your speed and accuracy.
Discuss your answers with your teachers or peers to get feedback and improve your understanding.

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Comments

ranchak singh hans
Mar 11, 2018

i am ranchak singh hans i hope you are well

dagani
Jun 20, 2016

i dint under stand