Question

If the 29th term of an AP is double of its 19th term, then find its 9th term.

Answer 1

Hint:We have to use the formula for having nth term of the AP. So, make two different expressions for its 29th term and 19th term. Then compare these two according to the question. Further find the suitable value for the expression of its 9th term.

Complete step-by-step answer:
As we know that the general term ${T_n}$ of an arithmetic progression (AP) is expressed as:
\[{T_n} = a + (n - 1)d\]
Where first term is ‘a’
And the common difference is ‘d’
Now, we have the expression for 29th term as follows,
\[
  {T_{29}} = a + (29 - 1)d \\
   \Rightarrow {T_{29}} = a + 28d…. (1) \\
 \]
Similarly, we have the expression for 19th term as follows,
\[
  {T_{19}} = a + (19 - 1)d \\
   \Rightarrow {T_{19}} = a + 18d …(2) \\
 \]
It is given that the 29th term is double of the 19th term of the AP.
So, from equations (1) and (2) , we have
${T_{29}} = 2 \times {T_{19}}$
Substituting the values in above we get,
\[
  a + 28d = 2 \times (a + 18d) \\
   \Rightarrow a + 28d = 2a + 36d \\
   \Rightarrow 2a - a = 28d - 36d \\
   \Rightarrow a = - 8d \\
 \]
Thus we got the relationship between ‘a’ and ‘d’ , as
$a = - 8d$
Doing algebraic transformation, we get
$a + 8d = 0 …(4)$
Now we know that general term of 9th term is $a+8d$,
So, from equation (4) we get,
${T_9} = 0 $
$\therefore $ The 9th term of the given AP will be 0.

Note:Arithmetic Progression or AP is a sequence of numbers, where the difference of any two consecutive numbers is constant. If we observe in our surroundings, then we may see it very easily. Remember that common difference and first term in AP will determine any particular term as well as the sum of it for any number of terms.