Question

For a G.P, r= 2 and S5 = 31, then first term a =​

Answer 1

$\begin{gathered}\begin{gathered}\bf \:Given-\begin{cases} &\sf{S_5 \: of \: G.P = 31} \\ &\sf{common \: ratio \: (r) = 2} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To \: Find - \begin{cases} &\sf{first \: term \: of \: G.P \: (a)}\end{cases}\end{gathered}\end{gathered}$

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

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of n terms of an geometric sequence is,

$\underline{ \boxed{ \bf \: S_n \: = \: a\bigg( \dfrac{ {r}^{n} - 1 }{r - 1} \bigg)}}$

Wʜᴇʀᴇ,

• Sₙ is the sum of first n terms.

• a is the first term of the sequence.

• n is the no. of terms.

• r is the common ratio.

Tʜᴜs,

↝ Sum of 5 terms is given by

$\rm :\longmapsto\:S_5 = a\bigg(\dfrac{ {r}^{5} - 1}{r - 1} \bigg)$

↝ It is given that Sum of first 5 terms is 31 and common ratio is 2,

Tʜᴜs,

$\rm :\longmapsto\:31 = a\bigg(\dfrac{ {2}^{5} - 1}{2 - 1} \bigg)$

$\rm :\longmapsto\:31 = a(32 - 1)$

$\rm :\longmapsto\:31 = 31a$

$\bf\implies \:a \: = \: 1$

$\overbrace{ \underline { \boxed { \bf \therefore \: The \: first \: term \: of \: G.P \: is \: 1}}}$

Additional Information :-

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an Geometric sequence is,

$\underline{ \boxed{ \bf \: a_n = {ar}^{n - 1}}}$

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.