Question

Find the term of terms in a gp​

Answer 1

Answer:

There are 7 terms in the GP

Step-by-step explanation:

In the geometric progression, there is a constant ratio between each two consecutive terms

The rule of the nth term of the geometric progression is

$a(n)=a(r)^{n-1}$, where

• a is the first term

• r is the constant ratio

• n is the position of the term in the sequence

∵ The first term = 5$\frac{1}{3}$

∴ a = 5$\frac{1}{3}$

∵ The last term = $\frac{243}{256}$

∴ a(n) = $\frac{243}{256}$

∵ The common ratio = $\frac{3}{4}$

→ Substitute these values in the rule above to find n

∵  $\frac{243}{256}$ =  5$\frac{1}{3}$ . $[\frac{3}{4}]^{n-1}$

→ Divide both sides by 5$\frac{1}{3}$

∴ $\frac{729}{4096}$ =  $[\frac{3}{4}]^{n-1}$

→ Let us find how many 3 in 729

∵ 729 ÷ 3 = 243 ÷ 3 = 81 ÷ 3 = 27 ÷ 3 = 9 ÷ 3 = 3 ÷ 3 = 1

∴ There are 6 three in 729

∴ 729 = $3^{6}$

→ Let us find how many 4 in 4096

∵ 4096 ÷ 4 = 1024 ÷ 4 = 256 ÷ 4 = 64 ÷ 4 = 16 ÷ 4 = 4 ÷ 4 = 1

∴ There are 6 four in 4096

∴ 4096 = $4^{6}$

∵  $\frac{729}{4096}$ = $\frac{3^{6} }{4^{6}}$ = $[\frac{3}{4}]^{6}$

∴  $[\frac{3}{4}]^{6}$ =  $[\frac{3}{4}]^{n-1}$

∵ The bases are equal

∴ Their exponents are equal

∴ 6 = n - 1

→ Add 1 to both sides

∴ 6 + 1 = n - 1 + 1

∴ 7 = n

∴ There are 7 terms in the GP