The eighth term of an A.p …

Let a be the first term and d be the common difference of the given A.P.
Then, 8^th term = a₈ = a + 7d
and 2^nd term = a₂ = a + d
According to given information,
{tex}a _ { 8 } = \frac { a _ { 2 } } { 2 }{/tex}
{tex}\Rightarrow{/tex} 2a₈ = a₂
{tex}\Rightarrow{/tex} 2(a + 7d) = a + d
{tex}\Rightarrow{/tex} a + 13d = 0...(i)
Now, 11^th term = a₁₁ = a + 10d and 4^th term = a₄ = a + 3d
According to the given information,
{tex}a _ { 11 } - \frac { a _ { 4 } } { 3 } = 1{/tex}
{tex}\Rightarrow{/tex} 3a₁₁ - a₄ = 3
{tex}\Rightarrow{/tex} 3(a + 10d) - (a + 3d) = 3
{tex}\Rightarrow{/tex} 3a + 30d - a - 3d = 3
{tex}\Rightarrow{/tex} 2a + 27d = 3...(ii)
Multiplying equation (i) by 2, we get
2a + 26d = 0...(iii)
Subtracting (iii) from (ii), we get
d = 3
{tex}\Rightarrow{/tex} a + 13(3) = 0
{tex}\Rightarrow{/tex} a = -39
Now, 15^th term = a_{​​​​​​15} = a + 14d = -39 + 14(3) = -39 + 42 = 3

Hence, 15^th term is 3 .