Recursive Formula Practice

Separate the one stated term from the equation containing a previous-term subscript.

The standalone stated term \(a_0=7\) fixes where the sequence begins. The remaining equation tells how later terms depend on earlier ones.

Separate the one stated term from the equation containing a previous-term subscript.

The standalone stated term \(b_1=-3\) fixes where the sequence begins. The remaining equation tells how later terms depend on earlier ones.

Separate the one stated term from the equation containing a previous-term subscript.

The standalone stated term \(c_4=10\) fixes where the sequence begins. The remaining equation tells how later terms depend on earlier ones.

Separate the one stated term from the equation containing a previous-term subscript.

The standalone stated term \(d_0=5\) fixes where the sequence begins. The remaining equation tells how later terms depend on earlier ones.

Separate the one stated term from the equation containing a previous-term subscript.

The standalone stated term \(p_2=12\) fixes where the sequence begins. The remaining equation tells how later terms depend on earlier ones.

Count the number of index steps from the given term to the requested term, updating the previous value each time.

Calculate every term in order: \(a_1=6+5=11,\ a_2=11+5=16\). Therefore \(a_2=16\).

Count the number of index steps from the given term to the requested term, updating the previous value each time.

Calculate every term in order: \(a_2=-4+7=3,\ a_3=3+7=10\). Therefore \(a_3=10\).

Count the number of index steps from the given term to the requested term, updating the previous value each time.

Calculate every term in order: \(a_1=12-3=9,\ a_2=9-3=6,\ a_3=6-3=3\). Therefore \(a_3=3\).

Count the number of index steps from the given term to the requested term, updating the previous value each time.

Calculate every term in order: \(a_3=9+4=13,\ a_4=13+4=17\). Therefore \(a_4=17\).

Count the number of index steps from the given term to the requested term, updating the previous value each time.

Calculate every term in order: \(a_2=15-6=9,\ a_3=9-6=3,\ a_4=3-6=-3\). Therefore \(a_4=-3\).

Mark each index above the term so a zero-based sequence does not lose or gain a multiplication.

There are \(3\) recursive steps. Multiplying the current term by \(2\) at each step gives \(u_3=24\).

Mark each index above the term so a zero-based sequence does not lose or gain a multiplication.

There are \(2\) recursive steps. Multiplying the current term by \(3\) at each step gives \(u_3=45\).

Mark each index above the term so a zero-based sequence does not lose or gain a multiplication.

There are \(3\) recursive steps. Multiplying the current term by \(0.5\) at each step gives \(u_3=8\).

Mark each index above the term so a zero-based sequence does not lose or gain a multiplication.

There are \(3\) recursive steps. Multiplying the current term by \(-2\) at each step gives \(u_5=16\).

Mark each index above the term so a zero-based sequence does not lose or gain a multiplication.

There are \(3\) recursive steps. Multiplying the current term by \(0.3333333333333333\) at each step gives \(u_4=3\).

Write the current n beside every substitution so the n-dependent term changes at the correct time.

Use both the current index and the immediately preceding value at each step: \(a_1=4,\ a_2=8,\ a_3=14\). Thus the requested term is \(14\).

Write the current n beside every substitution so the n-dependent term changes at the correct time.

Use both the current index and the immediately preceding value at each step: \(a_2=6,\ a_3=9,\ a_4=14\). Thus the requested term is \(14\).

Write the current n beside every substitution so the n-dependent term changes at the correct time.

Use both the current index and the immediately preceding value at each step: \(a_1=0,\ a_2=4,\ a_3=13\). Thus the requested term is \(13\).

Write the current n beside every substitution so the n-dependent term changes at the correct time.

Use both the current index and the immediately preceding value at each step: \(a_3=18,\ a_4=58,\ a_5=179\). Thus the requested term is \(179\).

Write the current n beside every substitution so the n-dependent term changes at the correct time.

Use both the current index and the immediately preceding value at each step: \(a_2=14,\ a_3=5,\ a_4=-7\). Thus the requested term is \(-7\).

Treat repeated composition as a recursion: record every intermediate function value.

The consecutive outputs are \(5,\ 9,\ 13,\ 17\). Each output becomes the next input, so the final result is \(17\).

Treat repeated composition as a recursion: record every intermediate function value.

The consecutive outputs are \(4,\ 10,\ 28\). Each output becomes the next input, so the final result is \(28\).

Treat repeated composition as a recursion: record every intermediate function value.

The consecutive outputs are \(2,\ 5,\ 26\). Each output becomes the next input, so the final result is \(26\).

Treat repeated composition as a recursion: record every intermediate function value.

The consecutive outputs are \(7,\ 3,\ 7,\ 3\). Each output becomes the next input, so the final result is \(3\).

Treat repeated composition as a recursion: record every intermediate function value.

The consecutive outputs are \(1,\ 5,\ 13,\ 29\). Each output becomes the next input, so the final result is \(29\).

Store the unrounded calculator result from each period and add the fixed contribution after multiplication.

Apply the recurrence 2 times, multiplying the previous balance before adding the contribution. Keeping full precision gives \(A_2=913.12\).

Store the unrounded calculator result from each period and add the fixed contribution after multiplication.

Apply the recurrence 3 times, multiplying the previous balance before adding the contribution. Keeping full precision gives \(A_3=1742.41\).

Store the unrounded calculator result from each period and add the fixed contribution after multiplication.

Apply the recurrence 2 times, multiplying the previous balance before adding the contribution. Keeping full precision gives \(A_2=2624.56\).

Store the unrounded calculator result from each period and add the fixed contribution after multiplication.

Apply the recurrence 3 times, multiplying the previous balance before adding the contribution. Keeping full precision gives \(A_3=732.91\).

Store the unrounded calculator result from each period and add the fixed contribution after multiplication.

Apply the recurrence 2 times, multiplying the previous balance before adding the contribution. Keeping full precision gives \(A_2=3457.92\).

Translate “retains, then adds” in that exact order; parentheses would be required for the reverse order.

The recurrence is \(I_n=0.8I_{n-1}+70\). Repeating it with each newly computed stock gives \(I_2=446\).

Translate “retains, then adds” in that exact order; parentheses would be required for the reverse order.

The recurrence is \(I_n=0.75I_{n-1}+30\). Repeating it with each newly computed stock gives \(I_3=170.63\).

Translate “retains, then adds” in that exact order; parentheses would be required for the reverse order.

The recurrence is \(I_n=0.9I_{n-1}+12\). Repeating it with each newly computed stock gives \(I_2=120\).

Translate “retains, then adds” in that exact order; parentheses would be required for the reverse order.

The recurrence is \(I_n=0.85I_{n-1}+45\). Repeating it with each newly computed stock gives \(I_3=361.41\).

Translate “retains, then adds” in that exact order; parentheses would be required for the reverse order.

The recurrence is \(I_n=0.6I_{n-1}+90\). Repeating it with each newly computed stock gives \(I_2=252\).

Use both neighboring rows when available: compute forward, then verify the following row.

Use the term immediately before the concealed row in \(a_n=2a_{n-1}+3\). Substitution produces \(a_2=25\), which also leads consistently to the next visible term.

Use both neighboring rows when available: compute forward, then verify the following row.

Use the term immediately before the concealed row in \(a_n=0.5a_{n-1}+2\). Substitution produces \(a_3=14.75\), which also leads consistently to the next visible term.

Use both neighboring rows when available: compute forward, then verify the following row.

Use the term immediately before the concealed row in \(a_n=3a_{n-1}+7\). Substitution produces \(a_1=1\), which also leads consistently to the next visible term.

Use both neighboring rows when available: compute forward, then verify the following row.

Use the term immediately before the concealed row in \(a_n=2a_{n-1}-2\). Substitution produces \(a_2=18\), which also leads consistently to the next visible term.

Use both neighboring rows when available: compute forward, then verify the following row.

Use the term immediately before the concealed row in \(a_n=0.8a_{n-1}+2\). Substitution produces \(a_1=82\), which also leads consistently to the next visible term.

Test a proposed rule on at least two transitions and verify its initial-condition index.

Substituting consecutive terms into \(a_1=3 and a_n=2a_{n-1}+2\) reproduces the full list and preserves the stated starting index. Each alternative fails either a transition or the index.

Test a proposed rule on at least two transitions and verify its initial-condition index.

Substituting consecutive terms into \(a_0=64 and a_n=0.5a_{n-1}+2\) reproduces the full list and preserves the stated starting index. Each alternative fails either a transition or the index.

Test a proposed rule on at least two transitions and verify its initial-condition index.

Substituting consecutive terms into \(a_2=5 and a_n=2a_{n-1}+3\) reproduces the full list and preserves the stated starting index. Each alternative fails either a transition or the index.

Test a proposed rule on at least two transitions and verify its initial-condition index.

Substituting consecutive terms into \(a_1=10 and a_n=a_{n-1}-3(n-1)\) reproduces the full list and preserves the stated starting index. Each alternative fails either a transition or the index.

Test a proposed rule on at least two transitions and verify its initial-condition index.

Substituting consecutive terms into \(a_0=2 and a_n=(n+2)a_{n-1}\) reproduces the full list and preserves the stated starting index. Each alternative fails either a transition or the index.

Audit the index, previous value, and operations as three separate checkpoints.

Each step must use the newly calculated previous term, not the initial value again. Recursive work must preserve the initial index, every intermediate state, and the operation order written in the rule.

Audit the index, previous value, and operations as three separate checkpoints.

The term immediately after a_0 is a_1, so the work has an index shift. Recursive work must preserve the initial index, every intermediate state, and the operation order written in the rule.

Audit the index, previous value, and operations as three separate checkpoints.

The formula multiplies the previous amount first and adds 50 afterward. Recursive work must preserve the initial index, every intermediate state, and the operation order written in the rule.

Audit the index, previous value, and operations as three separate checkpoints.

The current index n changes, so the squared addition must change with each term. Recursive work must preserve the initial index, every intermediate state, and the operation order written in the rule.

Audit the index, previous value, and operations as three separate checkpoints.

Intermediate rounding can accumulate error; retain precision until the requested final rounding. Recursive work must preserve the initial index, every intermediate state, and the operation order written in the rule.