Question

please help me in this question...
if for an AP a5=a10=5a..then find the value of a15 ...​

Answer 1

Given,

$\[\begin{array}{l}{a_5} = 5a\\{a_{10}} = 5a\end{array}\]$

Solution,

Consider the first term of A.P. is a and common difference is d.

$\[\begin{array}{l}a + \left( {5 - 1} \right)d = 5a\\ \Rightarrow a + 4d = 5a - - - - \left( 1 \right)\\a + \left( {10 - 1} \right)d = 5a\\ \Rightarrow a + 9d = 5a - - - - \left( 2 \right)\end{array}\]$

Calculate the value of $\[{a_{15}}\]$

$\[\begin{array}{l}{a_{15}} = a + \left( {15 - 1} \right)d\\ \Rightarrow {a_{15}} = a + 14d\\ \Rightarrow {a_{15}} = a + 4d + 10d\\ \Rightarrow {a_{15}} = 5a + 10 \times \left( {\frac{{5a - a}}{4}} \right)\\ \Rightarrow {a_{15}} = 5a + 10 \times \left( {\frac{{4a}}{4}} \right)\\ \Rightarrow {a_{15}} = 5a + 10 \times a\\ \Rightarrow {a_{15}} = 15a\end{array}\]$

Hence the value of $\[{a_{15}}\]$ is $\[15a\]$