the amount of money in nadias savings account at the end of $n$ years is represented by the recursive formula $a_n = 1.025(a_{n-1}) + 50$. if $a_6 = 291.51$, what is the balance in nadias account at the end of each of the first two years?

\$25.36, \$75.99
\$50.00, \$101.25
\$101.25, \$153.78
\$75.99, \$127.89

Question

the amount of money in nadias savings account at the end of \$n\$ years is represented by the recursive formula \$a_n = 1.025(a_{n-1}) + 50\$. if \$a_6 = 291.51\$, what is the balance in nadias account at the end of each of the first two years?

\\$25.36, \\$75.99
\\$50.00, \\$101.25
\\$101.25, \\$153.78
\\$75.99, \\$127.89

\$25.36, \$75.99
\$50.00, \$101.25
\$101.25, \$153.78
\$75.99, \$127.89

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Recursive Formulas" ], "new_concepts": [], "current_concepts": [ "Recursive Formulas", "Sequence Modeling" ] } ## Express the recursive relation backwards $$ \begin{aligned} a_n &= 1.025(a_{n-1}) + 50 \ a_{n-1} &= \frac{a_n - 50}{1.025} \end{aligned} $$ ## Calculate preceding terms from $a_6 = 291.51$ $$ \begin{aligned} a_5 &= \frac{291.51 - 50}{1.025} = 235.6195 \approx 235.62 \ a_4 &= \frac{235.6195 - 50}{1.025} = 181.0922 \approx 181.09 \ a_3 &= \frac{181.0922 - 50}{1.025} = 127.8948 \approx 127.89 \ a_2 &= \frac{127.8948 - 50}{1.025} = 75.9949 \approx 75.99 \ a_1 &= \frac{75.9949 - 50}{1.025} = 25.3609 \approx 25.36 \end{aligned} $$ ## Identify the balances for the first two years $$ \begin{aligned} a_1 &\approx \$25.36 \ a_2 &\approx \$75.99 \end{aligned} $$

(A) $25.36, $75.99

(B) $50.00, $101.25

(C) $101.25, $153.78

(D) $75.99, $127.89

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Recursive Formulas" ] }

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QUESTION IMAGE

Question

Explanation:

Express the recursive relation backwards

Calculate preceding terms from \(a_6 = 291.51\)

Identify the balances for the first two years

Answer: