The product of 5 terms of a gp is
243. the 3rd term of gpis equal
to 10th term of ap. find sum of
19th term in arthematic series.

The 19th term of Arithmetic series is - 2.02  

Step-by-step explanation:

Given as :

The product of 5 terms of a GP = 243

The 3rd term of GP is = 10th term of AP

Let The sum of 19th term of AP = x

Now, According to question

∵ nth term of GP is given by

= a×

where a is the first term and n is nth term  and r is the common ratio

So For n = 1

= a×

Or, = a×

i.e = a × 1

Or, = a

For n = 2

= a×

Or, = a×

i.e = a × r

For n =3

= a×

Or, = a×

i.e = a × r²

For n =4

= a×

Or, = a×

i.e = a × r³

For n =5

= a×

Or, = a×

i.e = a ×

Now, According to question

product of 5 terms of a GP = 243

So, × × × ×  = 243

Or, a × a r × a  r²× a  r³× a = 243

Or,   ×   =  

∴, a r² = 3                   ............1

Again

3rd term of GP = 10th term of AP

∵ nth term for AP

Tn = a + (n - 1) r  , where a is first term and r is common difference

So, 10th term of AP

= a + (10 - 1) r

= a + 9 r

∵ =

Or, a r² = a + 9 r

Now, from eq 1

3 = a + 9 r

i.e a + 9 r = 3

Or, + 9 r = 3

Or, 3 + 9 r³ = 3 r²

Or, 9 r³ - 3 r² + 3 = 0

Or, r = - 0.59

and a =  

i.e a = 8.6

Now,  19th term of AP

= a + (19 - 1) r

= 8.6 + 18 × (-.59)

∴   = - 2.02

So, The 19th term of Arithmetic series = tex]A_19[/tex] = - 2.02

Hence, The 19th term of Arithmetic series is - 2.02  Answer

answer4 to the second power is 16

and 4 to the third power is 64,