Math / Algebra

QuestionThe $n^{\text {th }}$ term of a sequence is given by $2 n+5$ Work out the first three terms of this sequence. a) How much does the sequence go up by between each term? b) Explain how you could have used the $n^{\text {th }}$ term rule to answer part a) without working out any of the terms.

Studdy Solution

STEP 1

1. The sequence is defined by the formula for the $n^{\text{th}}$ term: $a_n = 2n + 5$ .
2. We are tasked with finding the first three terms of the sequence.
3. We need to determine the difference between consecutive terms.
4. We need to explain how the $n^{\text{th}}$ term formula can be used to determine the difference without calculating individual terms.

STEP 2

1. Calculate the first three terms of the sequence.
2. Determine the difference between each consecutive term.
3. Explain the use of the $n^{\text{th}}$ term formula to find the difference.

STEP 3

Calculate the first term by substituting $n = 1$ into the formula:
$a_1 = 2(1) + 5 = 2 + 5 = 7$

STEP 4

Calculate the second term by substituting $n = 2$ into the formula:
$a_2 = 2(2) + 5 = 4 + 5 = 9$

STEP 5

Calculate the third term by substituting $n = 3$ into the formula:
$a_3 = 2(3) + 5 = 6 + 5 = 11$

STEP 6

Determine the difference between the first and second terms:
$a_2 - a_1 = 9 - 7 = 2$

STEP 7

Determine the difference between the second and third terms:
$a_3 - a_2 = 11 - 9 = 2$

STEP 8

Explain how the $n^{\text{th}}$ term formula can be used to find the difference:
The $n^{\text{th}}$ term formula is $a_n = 2n + 5$ . The coefficient of $n$ (which is 2) represents the constant difference between consecutive terms in an arithmetic sequence. Therefore, without calculating individual terms, we can determine that the sequence increases by 2 between each term.
The first three terms of the sequence are 7, 9, and 11. The sequence increases by 2 between each term. The $n^{\text{th}}$ term rule directly indicates this increase.

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QuestionThe $n^{\text {th }}$ term of a sequence is given by $2 n+5$ Work out the first three terms of this sequence. a) How much does the sequence go up by between each term? b) Explain how you could have used the $n^{\text {th }}$ term rule to answer part a) without working out any of the terms.

Studdy Solution

STEP 1

STEP 2

1. Calculate the first three terms of the sequence.
2. Determine the difference between each consecutive term.
3. Explain the use of the $n^{\text{th}}$ term formula to find the difference.

STEP 3

Calculate the first term by substituting $n = 1$ into the formula:
$a_1 = 2(1) + 5 = 2 + 5 = 7$

STEP 4

Calculate the second term by substituting $n = 2$ into the formula:
$a_2 = 2(2) + 5 = 4 + 5 = 9$

STEP 5

Calculate the third term by substituting $n = 3$ into the formula:
$a_3 = 2(3) + 5 = 6 + 5 = 11$

STEP 6

Determine the difference between the first and second terms:
$a_2 - a_1 = 9 - 7 = 2$

STEP 7

Determine the difference between the second and third terms:
$a_3 - a_2 = 11 - 9 = 2$

STEP 8

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Studdy Solution

STEP 1

STEP 2

1. Calculate the first three terms of the sequence.2. Determine the difference between each consecutive term.3. Explain the use of the nth n^{\text{th}} nth term formula to find the difference.

STEP 3

Calculate the first term by substituting n=1 n = 1 n=1 into the formula:a1=2(1)+5=2+5=7 a_1 = 2(1) + 5 = 2 + 5 = 7 a1​=2(1)+5=2+5=7

STEP 4

Calculate the second term by substituting n=2 n = 2 n=2 into the formula:a2=2(2)+5=4+5=9 a_2 = 2(2) + 5 = 4 + 5 = 9 a2​=2(2)+5=4+5=9

STEP 5

Calculate the third term by substituting n=3 n = 3 n=3 into the formula:a3=2(3)+5=6+5=11 a_3 = 2(3) + 5 = 6 + 5 = 11 a3​=2(3)+5=6+5=11

STEP 6

Determine the difference between the first and second terms:a2−a1=9−7=2 a_2 - a_1 = 9 - 7 = 2 a2​−a1​=9−7=2

STEP 7

Determine the difference between the second and third terms:a3−a2=11−9=2 a_3 - a_2 = 11 - 9 = 2 a3​−a2​=11−9=2

STEP 8

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1. Calculate the first three terms of the sequence.
2. Determine the difference between each consecutive term.
3. Explain the use of the $n^{\text{th}}$ term formula to find the difference.

Calculate the first term by substituting $n = 1$ into the formula:
$a_1 = 2(1) + 5 = 2 + 5 = 7$

Calculate the second term by substituting $n = 2$ into the formula:
$a_2 = 2(2) + 5 = 4 + 5 = 9$

Calculate the third term by substituting $n = 3$ into the formula:
$a_3 = 2(3) + 5 = 6 + 5 = 11$

Determine the difference between the first and second terms:
$a_2 - a_1 = 9 - 7 = 2$

Determine the difference between the second and third terms:
$a_3 - a_2 = 11 - 9 = 2$