Question

In an A.P. the 10^th term is 46 sum of the 5^th and 7^th term is 52. Find the A.P.

Answer 1

It is given that,
t₁₀ = 46
t₅ + t₇ = 52
Now,

\[t_n = a + \left( n - 1 \right)d\]

\[ t_{10} = a + \left( 10 - 1 \right)d\]

\[ \Rightarrow 46 = a + 9d\]

\[ \Rightarrow a = 46 - 9d . . . \left( 1 \right)\]

\[ t_5 + t_7 = 52\]

\[ \Rightarrow \left( a + \left( 5 - 1 \right)d \right) + \left( a + \left( 7 - 1 \right)d \right) = 52\]

\[ \Rightarrow a + 4d + a + 6d = 52\]

\[ \Rightarrow 2a + 10d = 52\]

\[ \Rightarrow 2\left( 46 - 9d \right) + 10d = 52 \left( \text { from }\left( 1 \right) \right)\]

\[ \Rightarrow 92 - 18d + 10d = 52\]

\[ \Rightarrow 92 - 8d = 52\]

\[ \Rightarrow 8d = 92 - 52\]

\[ \Rightarrow 8d = 40\]

\[ \Rightarrow d = 5\]

\[ \Rightarrow a = 46 - 9\left( 5 \right) \left( \text { from }\left( 1 \right) \right)\]

\[ \Rightarrow a = 46 - 45\]

\[ \Rightarrow a = 1\]

Hence, the given A.P. is 1, 6, 11, 16, ....