Respuesta :
Answer:
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
[tex]g_n[/tex] = a×[tex]r^{n - 1}[/tex]
where a is the first term and n is nth term and r is the common ratio
So For n = 1
[tex]g_1[/tex] = a×[tex]r^{1 - 1}[/tex]
Or, [tex]g_n[/tex] = a×[tex]r^{0}[/tex]
i.e [tex]g_n[/tex] = a × 1
Or, [tex]g_n[/tex] = a
For n = 2
[tex]g_2[/tex] = a×[tex]r^{2 - 1}[/tex]
Or, [tex]g_n[/tex] = a×[tex]r^{1}[/tex]
i.e [tex]g_n[/tex] = a × r
For n =3
[tex]g_3[/tex] = a×[tex]r^{3 - 1}[/tex]
Or, [tex]g_3[/tex] = a×[tex]r^{2}[/tex]
i.e [tex]g_3[/tex] = a × r²
For n =4
[tex]g_4[/tex] = a×[tex]r^{4 - 1}[/tex]
Or, [tex]g_4[/tex] = a×[tex]r^{3}[/tex]
i.e [tex]g_4[/tex] = a × r³
For n =5
[tex]g_5[/tex] = a×[tex]r^{5 - 1}[/tex]
Or, [tex]g_5[/tex] = a×[tex]r^{4}[/tex]
i.e [tex]g_5[/tex] = a × [tex]r^{4}[/tex]
Now, According to question
product of 5 terms of a GP = 243
So, [tex]g_1[/tex] × [tex]g_2[/tex] × [tex]g_3[/tex] × [tex]g_4[/tex] × [tex]g_5[/tex] = 243
Or, a × a r × a r²× a r³× a [tex]r^{4}[/tex] = 243
Or, [tex]a^{5}[/tex] × [tex]r^{10}[/tex] = [tex]3^{5}[/tex]
∴, 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
[tex]A_10[/tex] = a + (10 - 1) r
[tex]A_10[/tex] = a + 9 r
∵ [tex]g_3[/tex] = [tex]A_10[/tex]
Or, a r² = a + 9 r
Now, from eq 1
3 = a + 9 r
i.e a + 9 r = 3
Or, [tex]\dfrac{3}{r^{2} }[/tex] + 9 r = 3
Or, 3 + 9 r³ = 3 r²
Or, 9 r³ - 3 r² + 3 = 0
Or, r = - 0.59
and a = [tex]\dfrac{3}{(-0.59)^{2} }[/tex]
i.e a = 8.6
Now, 19th term of AP
[tex]A_19[/tex] = a + (19 - 1) r
[tex]A_19[/tex] = 8.6 + 18 × (-.59)
∴ [tex]A_19[/tex] = - 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
