Question

5th term of an arithmetic sequence is 17 and its 10th term is 32 .

a) What is its common difference ?

b) What is its first term ?

c) Find the position of 92 in this sequence ?​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{First\:term=5}}}$

$\green{\tt{\therefore{Common\:difference=3}}}$

$\green{\tt{\therefore{Position\:of\:92=30th\:term}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given : }} \\ \tt: \implies Fifth \: term( a_{5}) = 17 \\ \\ \tt: \implies Tenth \: term( a_{10}) = 32 \\ \\ \red{\underline \bold{To \: Find : }} \\ \tt: \implies Common \: difference(d)= ? \\ \\ \tt: \implies First \: term( a_{1} )= ? \\ \\ \tt: \implies Position \: of \:92= ?$

• According to given question :

$\bold{As \: we \: know \: that} \\ \tt: \implies a_{5} = 17 \\ \\ \tt: \implies a + 4d = 17 - - - - - (1) \\ \\ \bold{Similarly,} \\ \tt: \implies a_{10} = 32 \\ \\ \tt: \implies a + 9d = 32 - - - - - (2) \\ \\ \text{Subtracting \: (1) \: from \: (2)} \\ \tt: \implies 9d - 4d = 32 - 17 \\ \\ \tt: \implies 5d = 15 \\ \\ \tt: \implies d = \frac{15}{5} \\ \\ \green{\tt: \implies d = 3} \\ \\ \text{Putting \: value \: of \: d \: in \: (1)} \\ \tt: \implies a + 4 \times 3 = 17 \\ \\ \tt: \implies a + 12 = 17 \\ \\ \tt: \implies a = 17 - 12 \\ \\ \green{\tt: \implies a = 5} \\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies a_{n} = a + (n - 1)d \\ \\ \tt: \implies 92 = 5 + (n - 1) \times 3 \\ \\ \tt: \implies 92 - 5 = (n - 1) 3 \\ \\ \tt: \implies \frac{87}{3} = n - 1 \\ \\ \tt: \implies 29 = n - 1 \\ \\ \green{\tt: \implies n = 30th \: term}$

Answer 2

Given :-

• 5th term of AP = 17
• 10th term of AP = 32

Solution:-

Here, 5th term = a + 4d

⇒ a + 4d = 17 - - - (Eq.1 )

Again, 10th term = a + 9d

⇒ a + 9d = 32 - - - (Eq.2 )

Now, subtracting (Eq.1) from (Eq.2)

⇒ a + 9d - a - 4d = 32 - 17

⇒ 5d = 15

• Dividing both terms by 5

⇒ d = 3

So, a) Common difference (d) = 3(Ans.)

$\rule{200}{1}$

Now,

Putting value of d in (Eq.1):-

⇒ a + 4(3) = 17

⇒ a + 12 = 17

⇒ a = 17 - 12

⇒ a = 5

So, b) First term(a) = 5 (Ans.)

$\rule{200}{1}$

As we got a = 5 & d = 3

We know formula for nth terms:-

☛ An = a + (n - 1)d

⇒ 92 = 5 + (n - 1)3

⇒ 92 = 5 + 3n - 3

⇒ 92 = 3n + 2

⇒ 92 - 2 = 3n

⇒ 3n = 90

Dividing both terms by 3

⇒ n = 30

So,c)Position of 92 in AP=30th term(Ans.)