14. Sequences and series

Answer: $-43$

The $n$th term is $a + (n-1)d$ — the step is taken $6$ times, not $7$: $11 + 6 \times -9 = -43$.

Common mistakes, and the answer each one gives:

• Used $a + nd$, taking the step one time too many → $-52$
• Multiplied the first term by the step → $-99$

Answer: $\frac{1}{32}$

The $n$th term is $ar^{n-1}$, so the ratio is applied $5$ times: $1 \times \left(\frac{1}{2}\right)^{5} = \frac{1}{32}$.

Common mistakes, and the answer each one gives:

• Used $ar^n$, applying the ratio one time too many → $\frac{1}{64}$
• Added the ratio each time instead of multiplying → $\frac{7}{2}$

Answer: $259$

The last term is $10 + 6 \times 9 = 64$, and a sum of an arithmetic sequence is the number of terms times the average of the first and last: $7 \times \dfrac{10 + 64}{2} = 259$.

Common mistakes, and the answer each one gives:

• Multiplied the number of terms by the LAST term rather than the average → $448$
• Multiplied the number of terms by the FIRST term → $70$

Answer: $61$

The $n$th term is $a + (n-1)d$ — the step is taken $9$ times, not $10$: $-11 + 9 \times 8 = 61$.

Common mistakes, and the answer each one gives:

• Used $a + nd$, taking the step one time too many → $69$
• Multiplied the first term by the step → $-88$

Answer: $144$

The $n$th term is $ar^{n-1}$, so the ratio is applied $4$ times: $9 \times \left(2\right)^{4} = 144$.

Common mistakes, and the answer each one gives:

• Used $ar^n$, applying the ratio one time too many → $288$
• Added the ratio each time instead of multiplying → $17$

Answer: $69$

The last term is $-1 + 5 \times 5 = 24$, and a sum of an arithmetic sequence is the number of terms times the average of the first and last: $6 \times \dfrac{-1 + 24}{2} = 69$.

Common mistakes, and the answer each one gives:

• Multiplied the number of terms by the LAST term rather than the average → $144$
• Multiplied the number of terms by the FIRST term → $-6$

Answer: $92$

The $n$th term is $a + (n-1)d$ — the step is taken $10$ times, not $11$: $2 + 10 \times 9 = 92$.

Common mistakes, and the answer each one gives:

• Used $a + nd$, taking the step one time too many → $101$
• Multiplied the first term by the step → $18$

Answer: $27$

The $n$th term is $ar^{n-1}$, so the ratio is applied $3$ times: $1 \times \left(3\right)^{3} = 27$.

Common mistakes, and the answer each one gives:

• Used $ar^n$, applying the ratio one time too many → $81$
• Added the ratio each time instead of multiplying → $10$

Answer: $893$

The last term is $2 + 18 \times 5 = 92$, and a sum of an arithmetic sequence is the number of terms times the average of the first and last: $19 \times \dfrac{2 + 92}{2} = 893$.

Common mistakes, and the answer each one gives:

• Multiplied the number of terms by the LAST term rather than the average → $1748$
• Multiplied the number of terms by the FIRST term → $38$

Answer: $29$

The $n$th term is $a + (n-1)d$ — the step is taken $8$ times, not $9$: $-3 + 8 \times 4 = 29$.

Common mistakes, and the answer each one gives:

• Used $a + nd$, taking the step one time too many → $33$
• Multiplied the first term by the step → $-12$

Answer: $4374$

The $n$th term is $ar^{n-1}$, so the ratio is applied $6$ times: $6 \times \left(3\right)^{6} = 4374$.

Common mistakes, and the answer each one gives:

• Used $ar^n$, applying the ratio one time too many → $13122$
• Added the ratio each time instead of multiplying → $24$

Answer: $352$

The last term is $7 + 10 \times 5 = 57$, and a sum of an arithmetic sequence is the number of terms times the average of the first and last: $11 \times \dfrac{7 + 57}{2} = 352$.

Common mistakes, and the answer each one gives:

• Multiplied the number of terms by the LAST term rather than the average → $627$
• Multiplied the number of terms by the FIRST term → $77$

Answer: $97$

The $n$th term is $a + (n-1)d$ — the step is taken $29$ times, not $30$: $10 + 29 \times 3 = 97$.

Common mistakes, and the answer each one gives:

• Used $a + nd$, taking the step one time too many → $100$
• Multiplied the first term by the step → $30$

Answer: $-256$

The $n$th term is $ar^{n-1}$, so the ratio is applied $5$ times: $8 \times \left(-2\right)^{5} = -256$.

Common mistakes, and the answer each one gives:

• Used $ar^n$, applying the ratio one time too many → $512$
• Added the ratio each time instead of multiplying → $-2$

Answer: $511$

The last term is $4 + 13 \times 5 = 69$, and a sum of an arithmetic sequence is the number of terms times the average of the first and last: $14 \times \dfrac{4 + 69}{2} = 511$.

Common mistakes, and the answer each one gives:

• Multiplied the number of terms by the LAST term rather than the average → $966$
• Multiplied the number of terms by the FIRST term → $56$

Answer: $-113$

The $n$th term is $a + (n-1)d$ — the step is taken $15$ times, not $16$: $7 + 15 \times -8 = -113$.

Common mistakes, and the answer each one gives:

• Used $a + nd$, taking the step one time too many → $-121$
• Multiplied the first term by the step → $-56$

Answer: $64$

The $n$th term is $ar^{n-1}$, so the ratio is applied $3$ times: $8 \times \left(2\right)^{3} = 64$.

Common mistakes, and the answer each one gives:

• Used $ar^n$, applying the ratio one time too many → $128$
• Added the ratio each time instead of multiplying → $14$

Answer: $666$

The last term is $6 + 11 \times 9 = 105$, and a sum of an arithmetic sequence is the number of terms times the average of the first and last: $12 \times \dfrac{6 + 105}{2} = 666$.

Common mistakes, and the answer each one gives:

• Multiplied the number of terms by the LAST term rather than the average → $1260$
• Multiplied the number of terms by the FIRST term → $72$

Answer: $-158$

The $n$th term is $a + (n-1)d$ — the step is taken $22$ times, not $23$: $-4 + 22 \times -7 = -158$.

Common mistakes, and the answer each one gives:

• Used $a + nd$, taking the step one time too many → $-165$
• Multiplied the first term by the step → $28$

Answer: $48$

The $n$th term is $ar^{n-1}$, so the ratio is applied $3$ times: $6 \times \left(2\right)^{3} = 48$.

Common mistakes, and the answer each one gives:

• Used $ar^n$, applying the ratio one time too many → $96$
• Added the ratio each time instead of multiplying → $12$

Answer: $945$

The last term is $0 + 14 \times 9 = 126$, and a sum of an arithmetic sequence is the number of terms times the average of the first and last: $15 \times \dfrac{0 + 126}{2} = 945$.

Common mistakes, and the answer each one gives:

• Multiplied the number of terms by the LAST term rather than the average → $1890$
• Multiplied the number of terms by the FIRST term → $0$

Answer: $42$

The $n$th term is $a + (n-1)d$ — the step is taken $15$ times, not $16$: $-3 + 15 \times 3 = 42$.

Common mistakes, and the answer each one gives:

• Used $a + nd$, taking the step one time too many → $45$
• Multiplied the first term by the step → $-9$

Answer: $\frac{3}{64}$

The $n$th term is $ar^{n-1}$, so the ratio is applied $6$ times: $3 \times \left(\frac{1}{2}\right)^{6} = \frac{3}{64}$.

Common mistakes, and the answer each one gives:

• Used $ar^n$, applying the ratio one time too many → $\frac{3}{128}$
• Added the ratio each time instead of multiplying → $6$

Answer: $494$

The last term is $-10 + 12 \times 8 = 86$, and a sum of an arithmetic sequence is the number of terms times the average of the first and last: $13 \times \dfrac{-10 + 86}{2} = 494$.

Common mistakes, and the answer each one gives:

• Multiplied the number of terms by the LAST term rather than the average → $1118$
• Multiplied the number of terms by the FIRST term → $-130$

Answer: $158$

The $n$th term is $a + (n-1)d$ — the step is taken $26$ times, not $27$: $2 + 26 \times 6 = 158$.

Common mistakes, and the answer each one gives:

• Used $a + nd$, taking the step one time too many → $164$
• Multiplied the first term by the step → $12$

Answer: $13122$

The $n$th term is $ar^{n-1}$, so the ratio is applied $7$ times: $6 \times \left(3\right)^{7} = 13122$.

Common mistakes, and the answer each one gives:

• Used $ar^n$, applying the ratio one time too many → $39366$
• Added the ratio each time instead of multiplying → $27$

Answer: $338$

The last term is $-4 + 12 \times 5 = 56$, and a sum of an arithmetic sequence is the number of terms times the average of the first and last: $13 \times \dfrac{-4 + 56}{2} = 338$.

Common mistakes, and the answer each one gives:

• Multiplied the number of terms by the LAST term rather than the average → $728$
• Multiplied the number of terms by the FIRST term → $-52$

Answer: $90$

The $n$th term is $a + (n-1)d$ — the step is taken $13$ times, not $14$: $-1 + 13 \times 7 = 90$.

Common mistakes, and the answer each one gives:

• Used $a + nd$, taking the step one time too many → $97$
• Multiplied the first term by the step → $-7$

Answer: $972$

The $n$th term is $ar^{n-1}$, so the ratio is applied $5$ times: $4 \times \left(3\right)^{5} = 972$.

Common mistakes, and the answer each one gives:

• Used $ar^n$, applying the ratio one time too many → $2916$
• Added the ratio each time instead of multiplying → $19$

Answer: $1287$

The last term is $12 + 17 \times 7 = 131$, and a sum of an arithmetic sequence is the number of terms times the average of the first and last: $18 \times \dfrac{12 + 131}{2} = 1287$.

Common mistakes, and the answer each one gives:

• Multiplied the number of terms by the LAST term rather than the average → $2358$
• Multiplied the number of terms by the FIRST term → $216$