Question

The sum of the first eight terms of a GP whose nth term is 2.3" (n = 1,2,3....) is
O 19880
o 19860
O 19660

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Answer 1

Correct Question:

The sum of the first eight terms of a GP whose nth term is $2\cdot 3^n$ is,

a) 19880 b) 19860  c) 19660 d) 19680

The sum of the first eight terms of a GP whose nth term is  $2\cdot 3^n$ is 19680. (option d)

• It is given that $2\cdot 3^n$. Substitute 1 for n and simplify to find the first term of G.P.

        $a_1=2\cdot 3^1\\=6$

• Substitute 2 for n and simplify to find the second term of G.P.

       $a_2=2\cdot 3^2\\=2\cdot 9\\=18$

• Find the common ratio.

        $\frac{a_2}{a_1}=\frac{18}{6}\\ =3$

• So, the first term of the G.P is 6 and the common ratio is 3.
• The sum of the nth term of G.P is given by

        $S_n=\frac{a(r^n-1)}{r-1}$

• Substitute 18 for n, 6 for a, and 3 for r into the formula of sum and simplify.

        $S_8=\frac{6(3^8-1)}{3-1}\\=\frac{6(6561-1)}{2} \\=\frac{6(6560)}{2} \\=19680$

• Thus, the sum of the 8th term of the G.P is 19680.

Answer 2

Answer:

The sum of the first eight terms of a GP whose nth term is $2$ · $3^{n}$ (n = 1, 2, 3....) is 19680.

Step-by-step explanation:

1. As it is given $2$ · $3^{n}$ , we can substitute 1 in place of n to determine the first term of the G.P.

$a_{1}$ $=$ $2$ · $3^{1}$ $= 6$

2. Now we will substitute 2 in place of n and determine the second term of the G.P.

$a_{2}$ $= 2$ · $3^{2}$ $= 2$ · $9$ $= 18$

3. As our next step, we will find the common ratio,

$\frac{a_{2} }{a_{1} } = \frac{18}{6} = 3$

4. Hence, the first term in the G.P. is 6 and the common ratio is 3.

5. Sum of the nth term of G.P. is given by the formula,

$S_{n} = \frac{a(r^{n} -1) }{r -1 }$

6. Let us substitute 18 in place of n, 6 in place of a, and 3 in place of r.

We get,

$S_{8} = \frac{6(3^{8}-1) }{3-1} \\= \frac{6(6561-1)}{2} \\= \frac{6(6560)}{2} \\= 19680$

Therefore, the sum of the first eight terms of the G.P. is 19680.

Note: The correct question is:

The sum of the first eight terms of a G.P. whose nth term is $2$ · $3^{n}$ is,

(a) 19880

(b) 19860

(c) 19660

(d) 19680

OPTION (d) 19680 is the sum of the first eight terms of the G.P.

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