Question

the ratio of the 2nd term to the 6th term of an ap is 4:5 and the 10th term is 32. find the first term and the common difference. ​

Answer 1

Answer:

a = 20

d = 4 / 3

Step-by-step explanation:

Given, T₂ : T₆ = 4 : 5

           T₁₀ = 32

From first given information, we can frame a equation like this:

(a + d) / (a + 5d) = 4 / 5   [Because, Tn = a + (n - 1)d]

⇒ 5(a + d) = 4(a + 5d)   [Cross-multiplication]

⇒ 5a + 5d = 4a + 20d

⇒ a = 15d              ......(i).....

Now, let's use the second given information,

a + 9d = 32

Let's substitute the value of a from eqn.(i) in the above,

⇒ 15d + 9d = 32

⇒ 24d = 32

⇒ d = 32 / 24 = 4 / 3

Let's substitute the value of d in eqn.(i),

⇒ a = 15 * 4 / 3

⇒ a = 60 / 3

⇒ a = 20  

Answer 2

2nd term of AP is a+d

6th term of AP is a+5d

ratio between 2nd and 6th term is 4:5

so, 5×(a+d)=4×(a+5d)

5a-4a=20d-5d

a=15d............eq1

10th term is 32

a+9d=32

put value of eq 1

15d+9d=32

d=32/24

=4/3

=1¹/3

a=15×4/3

= 20

ANS : COMMON DIFFERENCE IS 4/3

FIRST TERM IS 20

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