If $10$ times the ${10^{th}}$term of an A.P. is equal to $15$ times the ${15^{th}}$ term, show that the ${25^{th}}$term of A.P. is zero.
Last updated date: 18th Mar 2023
•
Total views: 307.2k
•
Views today: 5.87k
Answer
307.2k+ views
Hint: Use general term of A.P. i.e, ${T_n} = a + (n - 1)d$.
We, know that the ${n^{th}}$term of an A.P. is given as:
${T_n} = a + (n - 1)d$
$\therefore {10^{th}}$term of A.P. will be:
$
\Rightarrow {T_{10}} = a + (10 - 1)d \\
\Rightarrow {T_{10}} = a + 9d \\
$
Similarly, ${15^{th}}$term will be:
$
\Rightarrow {T_{15}} = a + (15 - 1)d, \\
\Rightarrow {T_{15}} = a + 14d \\
$
Now, according to question:
$10{T_{10}} = 15{T_{15}}$
So, putting values of ${T_{10}}$and ${T_{15}}$from above, we’ll get:
$
\Rightarrow 10(a + 9d) = 15(a + 14d) \\
\Rightarrow 10a + 90d = 15a + 210d \\
\Rightarrow 5a + 120d = 0 \\
\Rightarrow a + 24d = 0 \\
$
And ${25^{th}}$term of A.P. will be:
$
\Rightarrow {T_{25}} = a + (25 - 1)d \\
\Rightarrow {T_{25}} = a + 24d \\
$
Putting the value $a + 24d = 0$ from above, we get:
$ \Rightarrow {T_{25}} = 0.$
Hence the ${25^{th}}$term of A.P. is zero.
Note: Since ${25^{th}}$ term of A.P. is zero, we can conclude that the sum of the first 49 terms of this A.P. is zero. In that case, the sum of the first 24 terms will be negative of the sum of the last 24 terms and ${25^{th}}$ term is already zero.
We, know that the ${n^{th}}$term of an A.P. is given as:
${T_n} = a + (n - 1)d$
$\therefore {10^{th}}$term of A.P. will be:
$
\Rightarrow {T_{10}} = a + (10 - 1)d \\
\Rightarrow {T_{10}} = a + 9d \\
$
Similarly, ${15^{th}}$term will be:
$
\Rightarrow {T_{15}} = a + (15 - 1)d, \\
\Rightarrow {T_{15}} = a + 14d \\
$
Now, according to question:
$10{T_{10}} = 15{T_{15}}$
So, putting values of ${T_{10}}$and ${T_{15}}$from above, we’ll get:
$
\Rightarrow 10(a + 9d) = 15(a + 14d) \\
\Rightarrow 10a + 90d = 15a + 210d \\
\Rightarrow 5a + 120d = 0 \\
\Rightarrow a + 24d = 0 \\
$
And ${25^{th}}$term of A.P. will be:
$
\Rightarrow {T_{25}} = a + (25 - 1)d \\
\Rightarrow {T_{25}} = a + 24d \\
$
Putting the value $a + 24d = 0$ from above, we get:
$ \Rightarrow {T_{25}} = 0.$
Hence the ${25^{th}}$term of A.P. is zero.
Note: Since ${25^{th}}$ term of A.P. is zero, we can conclude that the sum of the first 49 terms of this A.P. is zero. In that case, the sum of the first 24 terms will be negative of the sum of the last 24 terms and ${25^{th}}$ term is already zero.
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

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

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
