If a number is added to each term of a geometric sequence, does the series remain a geometric series. What about an arithmetic series.

The nth term of a geometric series has the form a*r^(n -
1) where a is the first term of the series and r is the common
ratio.

Assume a number N is added to each term of the
series. This gives the nth term as a*r^(n - 1) + N. This expression cannot be expressed
in a form a'*r'^(n - 1). Therefore, the resulting series is not a geometric
series.

The nth term of an arithmetic series is given by a
+ (n - 1)*d, where a is the first term and d is the common difference. If a number N is
added to each term of the arithmetic series we get a + (n - 1)*d +
N

We can write this as (a + N) + (n - 1)*d which is the nth
term of an arithmetic series that has the first term as a + N and the common difference
is d.

When a number is added to the terms of an arithmetic
series we get a new arithmetic series.

Adding
a number to the terms of a geometric series does not result in a geometric series but
add a number to the terms of an arithmetic series results in a new arithmetic
series.