Question

An Arithmetic Progression has the first term as 2 and the fifth term as 30.
A Geometric Progression has a common ratio of -0.5. The sum of the first two
terms of the Geometric Progression is the same as the second term of the Arithmetic
Progression. Find the first term of the Geometric Progression.​

Answer 1

The first term of Geometric Progression is 18  .

Step-by-step explanation:

Given as :

The first term of an Arithmetic progression = a = 2

The fifth term of an Arithmetic progression = $t_5$ = 30

∵ nth term of an A.P = $t_n$ = a + ( n - 1 ) d

So, for n = 5 ,               $t_5$ = 2 + ( 5 - 1 ) d

i.e 2 + ( 5 - 1 ) d = $t_5$ = 30

Or, 2 + 4 d = 30

Or,  4 d = 30 - 2

Or, 4 d = 28

∴        d = $\dfrac{28}{4}$

i.e      d = 7

Common difference of A.P = d = 7

So, Second term of A.P = $t_2$ = 2 + ( 2 - 1 ) × 7

Or,  $t_2$ = 2 + 7

∴     $t_2$ = 9

Second term of A.P = $t_2$  = 9

Again

Common ratio of Geometric progression = r = - 0.5

According to question

The sum of the first two  terms of the Geometric Progression is the same as the second term of the Arithmetic Progression.

Now,

nth term of G.P = $x_n$  =  x $r^{n-1}$

So, first term of G.P =  $x_1$  =  x $r^{1-1}$

i.e   $x_1$  =  x $r^{0}$

∴     $x_1$  =  x

Second term of G.P = $x_2$  =  x $r^{2-1}$

i.e    $x_2$  =  x $r$

∴     $x_2$  =  x ( - 0.5 )

A/Q

  $x_1$  + $x_2$  = $t_2$

i.e x +  x ( - 0.5 ) = 9

Or, x - 0.5 x = 9

∴  0.5 x = 9

Or,  x = $\dfrac{9}{0.5}$

Therefor  x = 18

So, The first term of G.P =   $x_1$  =  x  = 18

Hence, The first term of Geometric Progression is 18  . Answer