Question

There are 4 numbers such that first 3 of them form an AP and the last 3 of them form a GP. The sum of the first and third number is 2 and that of second and fourth is 26. what are these numbers?

Answer 1

♧♧HERE IS YOUR ANSWER♧♧

Let, the numbers are a, b, c and d.

Given that,

a, b and c form an AP, then

b - a = c - b

=> 2b = a + c .....(i)

and

b, c and d form a GP, then

c/b = d/c

=> c² = bd .....(ii)

Also, sum of a and c is 2, and sum of b and d is 26.

Then,

a + c = 2 .....(iii)

and

b + d = 26 .....(iv)

Using (i) and (iii), we eliminate (a + c) and we get :

2b = 2

=> b = 1

Putting b = 1 in (iv), we get :

1 + d = 26

=> d = 25

Putting b = 1 and d = 25 in (ii), we get :

c² = 1 × 25 = 25 = 5²

=> c = ± 5

Putting b = 1 and c = 5 in (i), we get :

2 × 1 = a + 5

=> a = - 3

Again, putting b = 1 and c = - 5 in (ii), we get :

2 × 1 = a - 5

=> a = 7

Therefore, we have a set of numbers as :

a = - 3, b = 1, c = 5, d = 25

and

a = 7, b = 1, c = - 5, d = 25

So, we can verify AP and GP conditions, given by (i) and (ii) respectively.

♧♧HOPE THIS HELPS YOU♧♧

Answer 2

Solution:

_____________________________________________________________


Given:

Statement 1 : There are 4 numbers ,...

let first number be a,
let second number be b,
let third number be c,
let forth number be d,

remaining part of statement 1 : first 3 numbers form AP and last 3 numbers form GP.

Statement 2 : The first 3 of the 4 numbers form AP,.
The sum of first and third is 2,.

which means,

a + c = 2,

as per AP,

a+(a+2d) = 2a + 2d = 2,.

As we know that first 3 numbers form AP,

so,
 
second term is a+d,.

∴$\frac{a+c}{2}$ =  b,.

and hence,.

2a + 2d = 2,

=> a + d = 1,.. b=1,.
___________________

Statement 3 :

b + d = 26

1 + d = 26

=> d = 25
__________________

As per statement 1,

remaining part,. last 3  form GP series,.

b is first term in G.P

and d is 3third & last term in GP series,.

a = 1,  ar² = 25,.

r² = 25,.

and so, r =5,.

As r = 5,.

c = br = 1 x 5 = 5,.

________________

As,

first term of the AP series,.

a + c = 2,

a + 5 = 2

a = 2 - 5

a = -3,.

__________

  Hence,

  The  4 numbers are,.

             =>    -3,1,5,25..

_____________________________________________________________

                                         Hope it Helps!!