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Megoldás - Arithmetic sequences

The common difference equals:

1500

1 500

The sum of the sequence equals:

11000

11 000

The explicit formula of this sequence is:

a_{n} = 500 + (n - 1) \cdot 1500

a_n=500+(n-1)*1500

The recursive formula of this sequence is:

a_{n} = a_{(n - 1)} + 1500

a_n=a_((n-1))+1500

The nth terms:

500, 2000, 3500, 5000, 6500, 8000, 9500...

500,2000,3500,5000,6500,8000,9500...

Lépések megtekintése Tudj meg többet a Tigerrel Próbáld ki ezt:

Lépésről lépésre magyarázat

1. Find the common difference

Find the common difference by subtracting any term in the sequence from the term that comes after it.

$a_{2} - a_{1} = 2000 - 500 = 1500$

$a_{3} - a_{2} = 3500 - 2000 = 1500$

$a_{4} - a_{3} = 5000 - 3500 = 1500$

The difference of the sequence is constant and equals the difference between two consecutive terms.
$d = 1500$

2. Find the sum

Calculate the sum of the sequence using the sum formula:

$S u m = (n (a_{1} + a_{n})) / 2$

$S u m = (n * (a_{1} + a_{n})) / 2$

Plug in the terms.

$S u m = (4 * (a_{1} + a_{n})) / 2$

$S u m = (4 * (500 + a_{n})) / 2$

$S u m = (4 * (500 + 5000)) / 2$

Simplify the expression.

$S u m = (4 * (500 + 5000)) / 2$

$S u m = (4 * 5500) / 2$

$S u m = \frac{22000}{2}$

$S u m = 11000$

The sum of this sequence is $11000$ .

This series corresponds to the following straight line $y = 1500 x + 500$

3. Find the explicit form

The formula for expressing arithmetic sequences in their explicit form is:
$a_{n} = a_{1} + (n - 1) \cdot d$

Plug in the terms.
$a_{1} = 500$ (this is the 1st term)
$d = 1500$ (this is the common difference)
$a_{n}$ (this is the nth term)
$n$ (this is the term position)

The explicit form of this arithmetic sequence is:

$a_{n} = 500 + (n - 1) \cdot 1500$

4. Find the recursive form

The formula for expressing arithmetic sequences in their recursive form is:
$a_{n} = a_{(1 - n)} + d$

Plug in the d term.
$d = 1500$ (this is the common difference)

The recursive form of this arithmetic sequence is:

$a_{n} = a_{(n - 1)} + 1500$

5. Find the nth element

$a_{1} = a_{1} + (n - 1) \cdot d = 500 + (1 - 1) \cdot 1500 = 500$

$a_{2} = a_{1} + (n - 1) \cdot d = 500 + (2 - 1) \cdot 1500 = 2000$

$a_{3} = a_{1} + (n - 1) \cdot d = 500 + (3 - 1) \cdot 1500 = 3500$

$a_{4} = a_{1} + (n - 1) \cdot d = 500 + (4 - 1) \cdot 1500 = 5000$

$a_{5} = a_{1} + (n - 1) \cdot d = 500 + (5 - 1) \cdot 1500 = 6500$

$a_{6} = a_{1} + (n - 1) \cdot d = 500 + (6 - 1) \cdot 1500 = 8000$

$a_{7} = a_{1} + (n - 1) \cdot d = 500 + (7 - 1) \cdot 1500 = 9500$

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Tudj meg többet a Tigerrel

When will the next bus arrive? How many people can fit inside a stadium? How much money will I earn this year? All these questions can be answered by learning how arithmetic sequences work. The progression of time, triangular patterns (bowling pins, for example), and increases or decreases in quantity can all be expressed as arithmetic sequences.

Fogalmak és témák

Arithmetic sequences