Question

the first term of two APs are equal and the ratios of their common differences is 1:2 . If the 7th term of first AP and the 21st term of second AP are 23 and 125 respectively. Find two APs​

Answer 1

given : a7=23,A2=125

A1:A2

D1:D2

A7 =23

A1+6D=23(1)

A21=125

a21+20d√2=125(2)

adding 1and2

20d√2_6d=102

but d1=1/2

d2

d2=2d

then

d1=d2/2=3/2

Answer 2

Answer:

Two APs are $5,8, 11,14, \ldots$and $5,11,17,23, \ldots$

Step-by-step explanation:

Let the first term be a

Common differences between the first and second APs be d1 and d2 respectively.

7th term = 23 and 21st term = 125

$\Rightarrow \mathrm{a}+6 \mathrm{~d}_{1}=23 \Rightarrow \mathrm{a}=23-6 \mathrm{~d}_{1} \ldots$ (i) and

$\mathrm{a}+20 \mathrm{~d}_{2}=125 \Rightarrow \mathrm{a}=125-20 \mathrm{~d}_{2} \ldots$ (ii)

From (i) and (ii) we get

$23-6 d_{1}=125-20 d_{2} \\ 20 d_{2}-6 d_{1}=102 \\ 10 d_{2}-3 d_{1}=51 \ldots (iii)$

But $d_{1}: d_{2}=1: 2$

$\Rightarrow d_{2}=2 d_{1}$

From (iii)

$20 d_{1}-3 d_{1}=51 \\ 17 d_{1}=51 \\ d_{1}=3 \\ d_{2}=6$

$\Rightarrow a+18=23 \Rightarrow a=5 .$ From (i)

Hence the two APs are $5,8, 11,14, \ldots$and $5,11,17,23, \ldots$