If the 9th term of an AP is zero, then prove that the 29th term is double of the 19th term.
Answer
383.4k+ views
Hint- Here, we will be using the formula for finding the nth term of an arithmetic progression in order to write the expressions of 9th term, 29th term and 19th term of an arithmetic progression.
To prove- 29th term is double of 19th term of given AP i.e., ${a_{29}} = 2{a_{19}}$
Let us suppose the first term of an AP as ${a_1}$, common difference as d.
As we know that the nth term of any AP with first term as ${a_1}$ and common difference as d is given by ${a_n} = {a_1} + \left( {n - 1} \right)d{\text{ }} \to {\text{(1)}}$
Given, ${a_9} = 0$
Using equation (1), we have
${a_9} = {a_1} + \left( {9 - 1} \right)d \Rightarrow {a_1} + 8d = 0{\text{ }} \to {\text{(2)}}$
Let us simplify the equation which needs to be proved by putting n=29 in equation (1) for LHS and n=19 in equation (1) for RHS, we can write
$
{a_{29}} = 2{a_{19}} \Rightarrow {a_1} + \left( {29 - 1} \right)d = 2\left[ {{a_1} + \left( {19 - 1} \right)d} \right] \Rightarrow {a_1} + 28d = 2\left( {{a_1} + 18d} \right) \Rightarrow {a_1} + 28d = 2{a_1} + 36d \\
\Rightarrow {a_1} + 8d = 0{\text{ }} \to {\text{(3)}} \\
$
Clearly, the equation which needs to be proved is reduced into equation (3) so in order to prove the required equation ${a_{29}} = 2{a_{19}}$, equation (3) needs to be proved.
Since, equation (2) holds true and equation (3) needs to be proved whereas equations (2) and (3) are the same. Hence, the required equation is proved i.e., ${a_{29}} = 2{a_{19}}$(29th term of AP is double of 19th term of AP).
Note- In these types of problems, we will simplify the equation that needs to be proved with the help of general formulas for an arithmetic progression and then use the already given condition. In this particular problem the given condition is the same equation (obtained after simplification) which needs to be proved.
To prove- 29th term is double of 19th term of given AP i.e., ${a_{29}} = 2{a_{19}}$
Let us suppose the first term of an AP as ${a_1}$, common difference as d.
As we know that the nth term of any AP with first term as ${a_1}$ and common difference as d is given by ${a_n} = {a_1} + \left( {n - 1} \right)d{\text{ }} \to {\text{(1)}}$
Given, ${a_9} = 0$
Using equation (1), we have
${a_9} = {a_1} + \left( {9 - 1} \right)d \Rightarrow {a_1} + 8d = 0{\text{ }} \to {\text{(2)}}$
Let us simplify the equation which needs to be proved by putting n=29 in equation (1) for LHS and n=19 in equation (1) for RHS, we can write
$
{a_{29}} = 2{a_{19}} \Rightarrow {a_1} + \left( {29 - 1} \right)d = 2\left[ {{a_1} + \left( {19 - 1} \right)d} \right] \Rightarrow {a_1} + 28d = 2\left( {{a_1} + 18d} \right) \Rightarrow {a_1} + 28d = 2{a_1} + 36d \\
\Rightarrow {a_1} + 8d = 0{\text{ }} \to {\text{(3)}} \\
$
Clearly, the equation which needs to be proved is reduced into equation (3) so in order to prove the required equation ${a_{29}} = 2{a_{19}}$, equation (3) needs to be proved.
Since, equation (2) holds true and equation (3) needs to be proved whereas equations (2) and (3) are the same. Hence, the required equation is proved i.e., ${a_{29}} = 2{a_{19}}$(29th term of AP is double of 19th term of AP).
Note- In these types of problems, we will simplify the equation that needs to be proved with the help of general formulas for an arithmetic progression and then use the already given condition. In this particular problem the given condition is the same equation (obtained after simplification) which needs to be proved.
Recently Updated Pages
Which of the following would not be a valid reason class 11 biology CBSE

What is meant by monosporic development of female class 11 biology CBSE

Draw labelled diagram of the following i Gram seed class 11 biology CBSE

Explain with the suitable examples the different types class 11 biology CBSE

How is pinnately compound leaf different from palmately class 11 biology CBSE

Match the following Column I Column I A Chlamydomonas class 11 biology CBSE

Trending doubts
Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

What is the past tense of read class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What is pollution? How many types of pollution? Define it

Give 10 examples for herbs , shrubs , climbers , creepers

Which state has the longest coastline in India A Tamil class 10 social science CBSE

Write an application to the principal requesting five class 10 english CBSE
