Series / Sequences: Complete the equation

To solve this problem, we'll follow these steps:

• Step 1: Identify the common pattern or rule in the sequence.
• Step 2: Use this pattern to determine the missing numbers.
• Step 3: Verify the pattern continues correctly with the numbers found.

Now, let's work through each step:
Step 1: The sequence provided is 20, 18, 16, 14, __, __, 8, 6, 4, 2.
We notice that each number differs from the previous one by $-2$ (20 to 18, 18 to 16, etc.). This suggests an arithmetic sequence with a common difference of $-2$.

Step 2: Let's continue this pattern to find the missing numbers. We have 16, 14, and then the blank spaces before reaching 8. So, following the pattern:
From 14, subtract 2 to get 12.
From 12, subtract 2 to get 10.
So, the missing numbers in the sequence are 12 and 10.

Step 3: Verify the sequence by checking the pattern:
Starting from 20: 20, 18, 16, 14, 12, 10, 8, 6, 4, 2.
Each step follows the pattern of subtracting 2.

Therefore, the missing numbers in the series are $12, 10$.

We start with the right column in the exercises.

Between each number there is a jump of +3:$4+3=7$

$7+3=10$

Etcetera.

Now we move to the left column of the exercises.

Between each number there is a jump of +1:

$2+1=3$

$3+1=4$

Now we can figure out which exercise is missing:

The left digit will be:$2-1=1$

The right digit will be:$4-3=1$

And the missing exercise is:$1+1$

To solve this problem, we will first identify the pattern in the sequence of additions and find the missing pair that correctly fits this pattern.

The given sequence is:

• $6+5$
• $5+4$
• $? + ?$
• $3+2$
• $2+1$

Notice that in each step, both numbers in the addition pair decrease by 1:

• From $6+5$ to $5+4$, the numbers decrease to $5$ and $4$ respectively.
• The next pair following this decrement pattern would be $4+3$.
• After this, the sequence continues with $3+2$.
• Thus, the missing pair is $4+3$.

Let's verify this with the choices given:

• Choice 4: $4+3$

This matches the decrement pattern established in the sequence.

Therefore, the correct answer is $4+3$.

To solve this problem, let's carefully determine the number of squares in each sequence element:

• Step 1: Identify the sequence pattern.
  Element 1: $1^2 = 1$ square.
  Element 2: $2^2 = 4$ squares.
  Element 3: $3^2 = 9$ squares.
• Step 2: Recognize the pattern as the sequence of perfect squares.
  For each element $n$, the number of squares is $n^2$.
• Step 3: Calculate the number of squares in the fourth element.
  Element 4: $4^2 = 16$ squares.

Thus, the fourth element in the sequence will have $16$ squares.

To solve the problem, we will analyze the sequence of element growth:

• Step 1: Identify the pattern in the sequence from the image.
  - Normally, a sequence of squares that increases in size might do so according to $n^2$ (a perfect square sequence).
  - Observing the sequence, we see that the first element has $1^2 = 1$ square, the second element has $2^2 = 4$ squares, the third has $3^2 = 9$ squares, and the fourth follows similarly.
• Step 2: Apply the identified pattern to compute the 5th element of the sequence.
  - When following the pattern $n^2$, the 5th element would naturally contain $5^2 = 25$ squares.

Thus, the number of squares in the 5th element of the sequence is $25$.

To solve this problem, we'll follow these steps:

• Identify the sequence pattern: it follows perfect squares.
• Associate each step with its corresponding perfect square number.
• Calculate the 6th element's number of squares using the formula $n^2$.

Now, let's work through each step:
Step 1: The sequence is defined by the perfect square numbers: $1^2, 2^2, 3^2, \ldots$.
Step 2: Each element in this sequence corresponds to an integer squared, revealing the sequence as $1, 4, 9, 16, 25, \ldots$.
Step 3: For the 6th element, calculate $6^2$:
$6^2 = 36$

Therefore, the solution to the problem is 36.

To solve the problem, follow these steps:

• Step 1: Identify the pattern of the sequence in terms of squares.
• Step 2: Verify that the sequence matches a particular formula or pattern, such as perfect squares.
• Step 3: Calculate the 7th term using the pattern identified.

Now, let's work through the solution:

Step 1: Observe and decipher the pattern governing the sequence of squares. From the SVG hint of squares, it suggests a pattern linked to square numbers.

Step 2: Assume that the pattern is the sequence of perfect squares:
1st element has $1^2 = 1$ square
2nd element has $2^2 = 4$ squares
3rd element has $3^2 = 9$ squares
This indicates a clear pattern of the nth element having $n^2$ squares.

Step 3: To find the 7th element, apply $n^2$ for $n=7$:
$7^2 = 49$

Therefore, the number of squares in the 7th element of the sequence is $\boxed{49}$.

It is apparent, that for each successive number, a square is added in length and one in width.

Hence, the rule using the variable n is:

$a(n)=n^2$

Therefore, the eighth term will be:

$n^2=8\times8=16$

We calculate by replacing$n=11$

$2\times11+2=$

First we solve the multiplication exercise and then we add 2:

$22+2=24$

To solve this problem, we'll follow these steps:

• Step 1: Understand the sequence rule and identify the variable $n$.
• Step 2: Substitute the value of $n$ into the sequence formula.
• Step 3: Calculate the value of the expression obtained.

Let's work through them:
Step 1: The sequence is defined by the rule $2(n+4)$. Here, $n$ is the position in the sequence since we are asked for the 9th element, $n = 9$.
Step 2: Substitute $n = 9$ into the expression, which gives us $2(9+4)$.
Step 3: Calculate $9 + 4$ to get 13. Then multiply by 2 to obtain $2 \times 13 = 26$.

Thus, the value of the 9th element in the sequence is $26$.

To determine the 7th element of the sequence defined by the rule $2(2n - 2)$, we follow these steps:

• Step 1: Identify the given formula for the sequence, which is $2(2n - 2)$.

• Step 2: Simplify this formula to its basic form. By expanding, we have: $2 \times (2n - 2) = 4n - 4.$

• Step 3: Substitute $n = 7$ into the simplified formula $4n - 4$.

• Step 4: Perform the calculation:
  $\begin{aligned} 4n - 4 &= 4 \times 7 - 4 \\ &= 28 - 4 \\ &= 24 \end{aligned}$

Thus, the 7th element of the sequence is $\boldsymbol{24}$.

We know that the first term of the series is 15.

From here we can easily write the entire series, until we see if we reach 1.  

15, 13, 11, 9, 7, 5, 3, 1

The number 1 is indeed an element of the series!

To solve this problem, we need to recognize the patterns for both boys and girls across the groups:

• For the boys: The sequence given is 5, 9, 17, 21, 25 for groups A, B, D, E, F respectively. Observing the differences: $9 - 5 = 4$, $17 - 9 = 8$, $21 - 17 = 4$, $25 - 21 = 4$. This means the sequence is increasing by alternating values of 4 and 8. Thus, the missing number for group C (after 9) should be $9 + 8 = 17$.
• For the girls: The sequence given is 25, 21, 13, 9, 5. Observing the differences: $21 - 25 = -4$, $13 - 21 = -8$, $9 - 13 = -4$, $5 - 9 = -4$. This means the sequence is decreasing by alternating values of -4 and -8. Thus, the missing number for group C (after 21) should be $21 - 8 = 13$.

Therefore, the values for group C are:

13, 17

To solve this problem, we'll analyze the given sequence's patterns:

First, let's observe the sequences:

$40 + 52$

$? + ?$

$26 + 40$

$19 + 34$

$12 + 28$

Let's determine any consistent changes for the first numbers $(40, 26, 19, 12)$ and the second numbers $(52, 40, 34, 28)$.

Notice for the first numbers:
$40$ becomes $26$ which decreases by $14$,
$26$ becomes $19$ which decreases by $7$,
$19$ becomes $12$ which also decreases by $7$.

For the second numbers:
$52$ becomes $40$ which decreases by $12$,
$40$ becomes $34$ again decreasing by $6$,
$34$ becomes $28$ decreasing by $6$.

The pattern therefore for the first number decreases by $7$ for the second and third pairs; the second number decreases by $6$.

We can conclude the missing sequence is:

• The first sequence element should be $33$ computed as $40 - 7 = 33$.
• The second sequence element should be $46$ computed as $52 - 6 = 46$.

Therefore, the correct answer is $33 + 46$, which matches choice 3.

In conclusion, the missing sequence element that fits the pattern is $33 + 46$.

To solve this problem, we need to address the rule given for the sequence:

• The expression $n - 0.5n$ simplifies to $0.5n$. This represents the rule for the sequence elements.
• By substituting $n = 15$ into the simplified expression, we can find the 15th element of the sequence:
• $0.5 \times 15 = 7.5$.
• Since the sequence outputs might be intended to be integer values, the closest proper integer representation of 7.5 in this context is 7 (assuming standard rounding and context interpretation based on answer choices).

Thus, by following the sequence rule, the 15th element of the sequence is approximately given as $7$.

Therefore, the solution to the problem is $7$.

To solve this problem, we'll follow these steps:

• Identify the formula for the sequence: $a_n = 4n - 2$.
• Substitute $n = 12$ into the formula to find the 12th element.
• Calculate the expression to determine the value of the 12th term.

Now, let's work through each step:

Step 1: Understand the sequence formula. The sequence is defined by $4n - 2$, where $n$ is the term number.

Step 2: Substitute the value of $n = 12$ into the formula.
Calculating, we have:

$a_{12} = 4 \times 12 - 2$

Step 3: Perform the calculation.
$a_{12} = 48 - 2 = 46$

Therefore, the 12th element of the sequence is $46$.

This matches choice 4, confirming our solution with the provided options.

The third term in the sequence is the term $a_3$:

$a_n= \frac{n}{2}$

We need to substitute in the position of the term in the sequence:

$n=3$

Now, using our values:

$a_{\underline{n}}= \frac{\underline{n}}{2} \\ n=\underline{3}\\ \downarrow\\ a_{\underline{3}}=\frac{\underline{3}}{2}$

Now we substitute the position of the term in the sequence (3) in place of n. The substitution is shown with an underline in the expression above.

Therefore, the correct answer is answer C.

The fourth and fifth terms in the sequence are the terms: $a_4,\hspace{4pt}a_5$meaning in the general term formula given:

$a_n=10n-9$we need to substitute the position (of the requested term in the sequence):

$n=4$$$for - $a_4$and-

$$$n=5$for-

$a_5$ Let's do this for the fourth term:

$a_{\underline{n}}= 10\underline{n}-9 \\ n=\underline{4}\\ \downarrow\\ a_{\underline{4}}= 10\cdot\underline{4}-9=40-9\\ a_4=31$when we substituted in place of n the position (of the requested term in the sequence): 4, substitution is shown with an underline in the expression above,

Similarly, for the fifth term, $a_5$we get:

$a_{\underline{5}}= 10\cdot\underline{5}-9=50-9\\ a_5=41$ which means that:

$a_4=31,\hspace{4pt}a_5=41$Therefore the correct answer is answer A.

In order to determine the first five terms in the sequence simply insert their positions into the given formula as shown below:

$a_n=3n+1$

We want to calculate the values of the terms:

$a_1,\hspace{4pt}a_2,\hspace{4pt}a_3,\hspace{4pt}a_4,\hspace{4pt}a_5$

Let's start with the first term in the sequence,

$a_n=3n+1$

We need to insert the position of whichever term that we want to find.

In this case we want to find the first term so we'll substitute as shown below:

$n=1$

Proceed to calculate:

$a_{\underline{n}}= 3\underline{n}+1 \\ n=\underline{1}\\ \downarrow\\ a_{\underline{1}}=3\cdot\underline{1}+1=4$

When we substituted the position in question in the place of n : the substitution is shown with an underline (as shown above),

Repeat this exact action for all the requested terms in the sequence, meaning for the second through fifth terms:

$a_{\underline{2}}=3\cdot\underline{2}+1=7 \\ a_{\underline{3}}=3\cdot\underline{3}+1=10 \\ a_{\underline{4}}=3\cdot\underline{4}+1=13 \\ a_{\underline{5}}=3\cdot\underline{5}+1=16 \\$For the second term $a_2$ we substituted:$n=2$ in to the formula:

$a_n=3n+1$

For the third term $a_3$ we again substituted:$n=3$ and so on for the rest of the requested terms,

To summarize, we determined that the first five terms:

$a_1,\hspace{4pt}a_2,\hspace{4pt}a_3,\hspace{4pt}a_4,\hspace{4pt}a_5$

in the given sequence, are:

$4,\hspace{4pt}7,\hspace{4pt}10,\hspace{4pt}13,\hspace{4pt}16$

Therefore, the correct answer is answer A.

Let's solve the problem step-by-step:

• Step 1: Calculate the common difference
  The difference between 22 and 15 is $22 - 15 = 7$.
  The difference between 29 and 22 is $29 - 22 = 7$.
  The difference between 43 and the next term should be 7, therefore it needs confirmation.

• Step 2: Determine the pattern
  Since the differences between the consecutive terms (15, 22, 29) and (unknown, 43, 50) are consistent, the sequence has a common difference of 7.

• Step 3: Calculate the missing terms using the common difference
  - The term after 29 is $29 + 7 = 36$.
  - After 43, there must be 50, validating that $43 + 7 = 50$.
  - The term after 50 can be found by adding 7 again: $50 + 7 = 57$.
  - Add 7 once more to find the final missing term: $57 + 7 = 64$.

Therefore, the missing numbers in the sequence are 36, 57, and 64. These fill in the blanks and maintain the arithmetic progression.

The correct set of missing numbers is 36, 57, 64