# Arithmetic Progression Properties:

#### Arithmetic Progression Property (1):

If a constant number is added or subtracted with each term of an arithmetic series then the resulting new series will also be an arithmetic series with the same common difference (d).

Example(1): Let an arithmetic series is, $$1, 3, 5, 7, 9, ......$$

by adding 2 with each term of the series. Then the new series, $$3, 5, 7, 9, 11, .....$$ The new series is also an arithmetic series and the common difference (d) of the new series is also 2, same as in the original series.

Example(2): Let an arithmetic series is, $$5, 7, 9, 11, ......$$

by subtracting 3 from each term of the series. Then the new series, $$2, 4, 6, 8,.....$$ The new series is also an arithmetic series and the common difference (d) of the new series is also 2, same as in the original series.

#### Arithmetic Progression Property (2):

If a constant number is divided or multiplied with each term of an arithmetic series then the resulting new series will also be an arithmetic series.

Example(1): Let an arithmetic series is, $$2, 4, 6, 8,.....$$

dividing by 2 with each term of the series. Then the new series, $$1, 2, 3, 4.....$$ The new series is also an arithmetic series.

Example(2): Let an arithmetic series is, $$1, 3, 5, 7,.....$$

multiplying by 3 with each term of the series. Then the new series, $$1 \times 3, \ 3 \times 3, \ 5 \times 3, \ 7 \times 3.....$$ $$3, 9, 15, 21,.....$$ The new series is also an arithmetic series.

#### Arithmetic Progression Property (3):

If consistent terms of the two arithmetic series are added or subtracted then the new resulting series will also be an arithmetic series.

Example(1): Let two arithmetic series are, $$1, 3, 5, 7, 9, ......$$ $$2, 4, 6, 8, 10,......$$

after adding the consistent terms of the above two series, the new series will be, $$(1 + 2), (3 + 4), (5 + 6), (7 + 8),....$$ $$3, 7, 11, 15, 19,.....$$ Here the new resulting series is also an arithmetic series with the common difference 4.

Example(2): Let two arithmetic series are, $$5, 7, 9, 11,......$$ $$2, 4, 6, 8,......$$

after subtracting the consistent terms of the above two above series, the new series will be, $$(5 - 2), (7 - 4), (9 - 6), (11 - 8),....$$ $$3, 3, 3, 3,.....$$ Here the new resulting series is also an arithmetic series with the common difference zero (0).

#### Arithmetic Progression Property (4):

If consistent terms of the two arithmetic series are divided or multiplied then the new resulting series will "not" be an arithmetic series.

Example(1): Let two arithmetic series are, $$1, 3, 5, 7, 9, ......$$ $$2, 4, 6, 8, 10,......$$

after dividing the consistent terms of the above two series, the new series will be, $$\frac{1}{2}, \frac{3}{4}, \frac{5}{6}, \frac{7}{8}, \frac{9}{10}....$$ Here the new resulting series is not an arithmetic series.

Example(2): Let two arithmetic series are, $$1, 3, 5, 7, 9, ......$$ $$2, 4, 6, 8, 10,......$$

after multiplying the consistent terms of the above two series, the new series will be, $$1 \times 2, \ 3 \times 4, \ 5 \times 6, \ 7 \times 8.....$$ $$2, 12, 30, 56,.....$$ The new series is not an arithmetic series.