Is this the recursive formula for geometric sequences? {a1=first terman=r×an−1

We have {a1=first terman=r×an−1The second term of this sequence isa2=r×a2−1a2=r×a1 (2)The third term:a3=r×a3−1a3=r×a2Put value of a2 from equation 2⇒a3=r×(r×a1)⇒a3=r2×a1(3)Similarly, you can calculate a4a4=r×a4−1a4=r×a3Put a value of a3 from the equation (3)⇒a4=r×(r2×a1)⇒a4=r3×a1 (4)And so on. Thus, we have the followinga1,r×a1,r2×a1,....So we have observed thata2a1=a3a2=a4a3=...=anan−1=r Thus, it's the formula of geometric sequences.

Answered: "Is this the recursive formula for geometric sequences?..."

keche0b

Answered question

2022-01-04

Is this the recursive formula for geometric sequences?
${\begin{cases} a_{1} = f i r s t t e r m \\ a_{n} = r \times a_{n - 1} \end{cases}$

Answer & Explanation

Vivian Soares

Beginner2022-01-05Added 36 answers

We have
${\begin{cases} a_{1} = f i r s t t e r m \\ a_{n} = r \times a_{n - 1} \end{cases} The second term of this sequence is a_{2} = r \times a_{2 - 1} a_{2} = r \times a_{1} (2) The third term: a_{3} = r \times a_{3 - 1} a_{3} = r \times a_{2} Put value of a_{2} from equation 2 \Rightarrow a_{3} = r \times (r \times a_{1}) \Rightarrow a_{3} = r^{2} \times a_{1} (3) Similarly, you can calculate a_{4} a_{4} = r \times a_{4 - 1} a_{4} = r \times a_{3} Put a value of a_{3} from the equation (3) \Rightarrow a_{4} = r \times (r^{2} \times a_{1}) \Rightarrow a_{4} = r^{3} \times a_{1} (4) And so on. Thus, we have the following a_{1}, r \times a_{1}, r^{2} \times a_{1}, . . . . So we have observed that \frac{a_{2}}{a_{1}} = \frac{a_{3}}{a_{2}} = \frac{a_{4}}{a_{3}} = . . . = \frac{a_{n}}{a_{n - 1}} = r$
Thus, it's the formula of geometric sequences.

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