Question

$\huge\red{\boxed{\sf QuEsTIoN}}$

The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is​

Answer 1

Answer:

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Answer 2

$\large\underline{\sf{Solution-}}$

Given that,

• The first two terms of a geometric progression add up to 12.

• The sum of the third and the fourth terms is 48.

• The terms of the geometric progression are alternately positive and negative.

Since,

The terms of the geometric progression are alternately positive and negative, so it means common ratio < 0.

Let assume that

• First term of GP series is a

• Common ratio of GP series is r ( < 0).

According to first condition

The first two terms of a geometric progression add up to 12.

$\rm :\longmapsto\:a_1 + a_2 = 12$

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an geometric sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\: {r}^{n \: - \: 1} \:}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

• r is the common ratio.

Tʜᴜs,

$\rm :\longmapsto\:a + ar = 12$

$\bf\implies \:a(1 + r) = 12 - - - - (1)$

According to second condition

$\rm :\longmapsto\:a_3 + a_4 = 48$

$\rm :\longmapsto\: {ar}^{2} + {ar}^{3} = 48$

$\bf\implies \: {ar}^{2}(1 + r) = 48 - - - - (2)$

On dividing equation (2) by equation (1), we get

$\rm :\longmapsto\:\dfrac{ {ar}^{2} (1 + r)}{a(1 + r)} = \dfrac{48}{12}$

$\rm \implies\: {r}^{2} = 4$

$\rm \implies\:r \: = \: \pm \: 2$

As common ratio, r < 0.

$\bf \implies\:r \: = \: - \: 2 - - - - (3)$

On substituting the value of r in equation (1), we get

$\rm :\longmapsto\:a(1 - 2) = 12$

$\rm :\longmapsto\:a(- 1) = 12$

$\bf\implies \:a \: = \: - \: 12$

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Additional Information :-

↝ Sum of n  terms of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:= \frac{a \: ( \: {r}^{n} \: - \: 1 \: )}{r \: - \: 1} \: \: and \: \: r \: \ne \: 1 \: }}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• Sₙ is the sum of n terms of AP.

• a is the first term of the sequence.

• n is the no. of terms.

• r is the common ratio.