Question

the ratio of the sum of n terms of twi AP is (7n+1):(4n+27)find the ratio of their 15th term​

Answer 1

Answer:

$\large{\underline{\boxed{\sf \dfrac{T_n}{T'_n} = \dfrac{204}{143}}}}$

Step-by-step explanation:

Let a₁ and a₂ be the first terms and d₁ and d₂ be the common difference of the two APₛ respectively.

Let Sₙ and Sʼₙ be the sums of the first n terms of the two APₛ and Tₙ and Tʼₙ be their nth term respectively.

Then,

$\sf \frac{S_n}{S'_n} = \frac{7n + 1}{4n + 27} \\ \\ \implies \sf \frac{ \frac{n}{2} [2a_1 + (n - 1)d_1]}{ \frac{n}{2} [2a_2 + (n - 1)d_2]} = \frac{7n + 1}{4n + 27} \\ \\ \implies \sf \frac{[2a_1 + (n - 1)d_1]}{[2a_2 + (n - 1)d_2]} = \frac{7n + 1}{4n + 27} \: ....(i)$

To find the ratio of mth term, we replace n by (2m - 1) in the above expression.

Replacing n by (2 * 15 - 1), i.e., 29 on bpth sides in (i), we get -

$\implies \sf \dfrac{[2a_1 + (29 - 1)d_1]}{[2a_2 + (29 - 1)d_2]} = \dfrac{7(29) + 1}{4(29) + 27}$

$\implies \sf \frac{[2a_1 +28d_1]}{[2a_2+ 28d_2]} = \frac{204}{143} \\ \\ \implies \sf \frac{(a_1 + 14d_1)}{(a_2 + 14d_2)} = \frac{204}{143} \\ \\ \implies \sf \frac{a_1 + (15 - 1)d_1}{a_2 + (15 - 1)d_2} = \frac{204}{143} \\ \\ \implies \sf \frac{T_n}{T'_n} = \frac{204}{143}$

∴ Required Ratio = 204 : 143.

Answer 2

$\huge\bold\green{Answer:-}$

Given ratio of sum of n terms of two AP’s

= (7n+1):(4n+27)

We can consider the nth term as the m th term.

Let’s consider the ratio these two AP’s m th terms as am : a’m →(2)

Recall the nth term of AP formula, an = a + (n – 1)d

Hence equation (2) becomes,

am : a’m = a + (m – 1)d : a’ + (m – 1)d’

On multiplying by 2, we get

am : a’m = [2a + 2(m – 1)d] : [2a’ + 2(m – 1)d’]

= [2a + {(2m – 1) – 1}d] : [2a’ + {(2m – 1) – 1}d’]

= S2m – 1 : S’2m – 1 

= [7(2m – 1) + 1] : [4(2m – 1) +27] [from (1)]

= [14m – 7 +1] : [8m – 4 + 27]

= [14m – 6] : [8m + 23]

Thus the ratio of mth terms of two AP’s is [14m – 6] : [8m + 23].

Now,substitute the value of m as 15

We get,

[14×15-6] : [8×15+23]

[204] : [143]

=$\sf\dfrac{204}{143}$

= 204:143