Question

Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.​

Answer 1

Answer:

$\huge{\pink{\boxed{\green{\sf{a31=178}}}}}$

Step-by-step explanation:

$\huge{\pink{\boxed{\green{\underline{\red{\sf{SOLUTION-}}}}}}}$

$\to a11 = 38 \\ \to a + 10d = 38 - - - - - (1) \\ \\ \to a16 = 73 \\ \to a + 15d = 73 - - - - - (2)$

☆ According to question

☆ We get two eqn two unknown so subtract (1) from (2)

$\to a + 15d - (a + 10d) = 73 - 38 \\ \to a + 15d - a - 10d = 35 \\ \to 5d = 35 \\ \to d = \frac{35}{5} \\ \to d = 7 \\ \\ putting \: value \: of \: d \: in \: (1) \\ \to a + 10d = 38 \\ \to a + 10 \times 7 = 38 \\ \to a + 70 = 38 \\ \to a = 38 - 70 \\ \to a = - 32 \\ \\ \to a31 = a + 30d \\ \to a31 = - 32 + 30 \times 7 \\ \to a31 = - 32 + 210 \\ \to a31 = 178$

$\huge{\pink{\boxed{\green{\sf{a31=178}}}}}$

_________________________________________

Answer 2

$\boxed{\tt \dagger Given :- \dagger}$

11th term of AP is 38 and,

16th term of AP is 73.

$\boxed{ \tt \dagger To find :- \dagger}$

The 31st term of AP = ?

$\boxed{ \tt \dagger Solution :- \dagger}$

Let first term of AP be a

Let first term of AP be aand common difference be d

Let first term of AP be aand common difference be dNow,

$\tt \red{a_{11}=38a}$

$\tt\longrightarrow \green{a+10d=38\:.............(i)}$

And,

$\blue{\tt\:a_{16}=73a}$

$\tt\longrightarrow\pink{a+15d=73\:.............(ii)}$

From eq (i) and eq (ii),

a + 10d = 38 ‿︵‿︵│

⠀ ⠀ ⠀ ⠀⠀⠀ ⠀ ⠀ ⠀⠀⠀ ⠀ |Subtracting

$\boxed{+}$a $\boxed{+}$ 15d = $\boxed{+}$73 ‿︵‿︵│

-⠀ -⠀ ⠀ -

━━━━━━━━━━━━━━

-5d = -35

$\purple{ \tt⤇ d = \dfrac{-35}{-5} }$

$\orange{ \tt⤇ d = 7}$

Now,

Substitute the value of d in equation (i),

$\tt a + 10d = 38 \\ \tt⤇ a + 10 × 7 = 38 \\ \tt⤇ a + 70 = 38 \\ \tt⤇ a = 38 - 70 \\ \tt⤇ a = -32$

Then,

$\gray{\tt\:a_{31}=a+30da }$

$\tt\longrightarrow\:a_{31}=-32+30\times{7} \\ \tt\longrightarrow\:a_{31}=-32+210 \\ \tt\longrightarrow\:a_{31}=178$

Hence, the 31st term of an AP was $\boxed{\sf\pink{178.}}$