Find the next number in the sequence: 1, 5, 11, 19, 28, __

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7. Find the next number in the sequence below. 1, 5, 11, 19, 28, __

Find the differences between consecutive terms.
$5 - 1 = 4$
$11 - 5 = 6$
$19 - 11 = 8$
$28 - 19 = 9$
The sequence of first differences is 4, 6, 8, 9.
Find the differences between these first differences (second differences).
$6 - 4 = 2$
$8 - 6 = 2$
$9 - 8 = 1$
The sequence of second differences is 2, 2, 1.
Identify the pattern in the second differences.
The increment is decreasing by 1 each time (from +2 to +1). Assuming this pattern continues, the next second difference would be $1 - 1 = 0$ .
Use the next second difference to find the next first difference.
The last first difference was 9. Adding the next second difference (0) gives $9 + 0 = 9$ . So, the next first difference is 9.
Add the next first difference to the last term in the sequence to find the next number.
The last term was 28. Adding the next first difference (9) gives $28 + 9 = 37$ .
The next number in the sequence is $\boxed{37}$ .

8. Express 7800 in scientific notation.
Move the decimal point to the left until there is only one non-zero digit to its left.
For 7800, the decimal point is after the last zero. Move it 3 places to the left to get 7.8.
Count the number of places the decimal point was moved.
This number will be the exponent of 10. Since it was moved 3 places to the left, the exponent is 3.
$7800 = 7.8 \times 10^3$
In scientific notation, 7800 is $\boxed{7.8 \times 10^3}$ .

9. Expand 9658 using exponents.
Break down the number into its place values.
9658 consists of 9 thousands, 6 hundreds, 5 tens, and 8 ones.
Express each place value using powers of 10.
$9658 = (9 \times 1000) + (6 \times 100) + (5 \times 10) + (8 \times 1)$ $9658 = (9 \times 10^3) + (6 \times 10^2) + (5 \times 10^1) + (8 \times 10^0)$
The expanded form using exponents is $\boxed

Find the next number in the sequence: 1, 5, 11, 19, 28, __

Step-by-step solution

Find the differences between consecutive terms.

Find the differences between these first differences (second differences).

Identify the pattern in the second differences.

Use the next second difference to find the next first difference.

Add the next first difference to the last term in the sequence to find the next number.

Move the decimal point to the left until there is only one non-zero digit to its left.

Count the number of places the decimal point was moved.

Break down the number into its place values.

Express each place value using powers of 10.

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