Question

If the sum of the first 100 terms of a G.P is O and the
first term is -1, then the common ratio of the G.P will be:
A) 2
B) 1
C) 0
D)-1​

Answer 1

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

Given that

• Sum of first 100 terns of GP series = 0

• First term of GP series, a = - 1

• Let common ratio of GP series be 'r'.

We know that,

↝ Sum of n terms of an geometric sequence is,

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

Wʜᴇʀᴇ,

• Sₙ is the sum of n terms of GP.

• a is the first term of the sequence.

• n is the no. of terms.

• r is the common ratio.

Now, on substituting the values, we get

$\rm :\longmapsto\: - 1 \times \dfrac{ {r}^{100} - 1}{r - 1} = 0$

$\rm :\longmapsto\: \dfrac{ {r}^{100} - 1}{r - 1} = 0$

$\rm :\longmapsto\: {r}^{100} - 1 = 0$

$\rm :\longmapsto\: {r}^{100} = 1$

$\bf\implies \:r \: = - \: 1 \: \: or \: \: 1 \: \{rejected \}$

Hence,

$\bf\implies \:r \: = \: - \: 1$

$\: \: \: \: \: \: \: \: \: \: \: \: \underbrace{ \boxed{ \bf \: Option \: D \: is \: correct}}$

Reason why r = 1 is rejected.

We have,

First term, a = - 1

Common ratio, r = 1

So, series become,

-1, - 1, - 1, - 1, ------, - 1 (100 th term) which never adds up to give 0.

So r = 1 is rejected.

Additional Information :-

↝ 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 terms of GP.

• a is the first term of the sequence.

• n is the no. of terms.

• r is the common ratio