Question

If sum of the n terms of two AP are in ratio 5n+4: 9n+6 find the ratio of their 25th terms

Answer 1

For first A.P.

T = a

Common difference = d

Sum = n/2(2a+(n-1)d)

n Term = a+(n-1)d

For second A.P.

T = A

Common difference = F

Sum = n/2(2A+(n-1)D)

n Term = A+(n-1)D

Given,

$\frac {n/2(2a+(n-1)d} {n/2(2A+(n-1)D)}$ =

$\frac {5n+4}{9n+6}$ 

Ratio of 25th term,

$\frac {a+24d}{A+24D}$

We need to convert this ratio in the given ratio;

Divide given ratio by 2;

We get

[tex} \frac {a+((n-1)/2)d} {A+((n-1)/2)D} [\tex] = 5n+4/9n+6


Comparing it with requried ratio we find (n-1)/2=24

n-1 = 48

n=49

Put the value

5n+4:9n+6

5(49)+4 : 9(49)+6

249:447

83:149

ANSWER IS 83:149

Answer 2

an = a1 + (n - 1)d

Find the 25th term of the AP1:

a25 = a + (25 - 1)d

a25 = a + 24d

Find the 25th term of AP2:

*We will denote a1 as "A" and d as "D" to differentiate between the 2 APs

A25 = A + (25 - 1)D

A25 = A + 24D

Express the ratio of the two 25th term2:

$\text{Ratio = }\dfrac{a + 24d}{A + 24D}$

Express the sum of the 25 terms in AP1 in term of a and d:

Sn = n/2 (2a + (n - 1)d)

$S_{25} = \dfrac{n}{2} (2a + (n - 1)d)$

Express the sum of the 25 terms in AP2 in term of A and D:

Sn = n/2 (2a + (n - 1)d)

$S_{25} = \dfrac{n}{2} (2A + (n - 1)D)$

Form their ratio of their 25 terms:

$\text {Ratio = } \dfrac{n}{2} (2a + (n - 1)d) : \dfrac{n}{2} (2A + (n - 1)D)$

Divide by n/2 on both sides:

$\text {Ratio = } 2a + (n - 1)d : } 2A + (n - 1)D$

Divide by 2 on both sides:

$\text {Ratio = } a + \dfrac{(n - 1)d }{2} : } A +\dfrac{ (n - 1)D}{2}$

Form the equation:

Ratio of sum of 25 terms of AP1 : AP2 = 5n + 4 : 9n + 6

$\dfrac{a + \dfrac{(n - 1)d }{2}}{A +\dfrac{ (n - 1)D}{2} }= \dfrac{5n + 4}{9n + 6}$

Comparing it with the 25th term ratio that we are supposed to find:

$\text{Ratio = }\dfrac{a + 24d}{A + 24D}$

We can observe that:

$\dfrac{n-1}{2}= 24$

$n- 1 = 48$

$n = 49$

Substitute n = 49 into the ratio:

$\dfrac{a + \dfrac{(n - 1)d }{2}}{A +\dfrac{ (n - 1)D}{2} }= \dfrac{5n + 4}{9n + 6}$

$\dfrac{a + \dfrac{(49 - 1)d }{2}}{A +\dfrac{ (49 - 1)D}{2} }= \dfrac{5(49) + 4}{9(49 + 6}$

$\dfrac{a + 24d}{A + 24D} = \dfrac{5(49) + 4}{9(49 + 6}$

$\dfrac{a + 24d}{A + 24D} = \dfrac{249}{447} = \dfrac{83}{149}$

Answer: The ratio is 83 : 149