Let's denote the first term of the geometric sequence as \(\( a \)\) and the common ratio as \(\( r \)\). In such a sequence, the second term would be \(\( ar \)\) and the third term would be \(\( ar^2 \)\).
According to the problem:
- When 42 is added to the first term \(\( a \)\), we get \(\( a + 42 \)\).
- When 32 is added to the second term \(\( ar \)\), we get \(\( ar + 32 \)\).
- When 2 is added to the third term \(\( ar^2 \)\), we get \(\( ar^2 + 2 \)\).
All these three modified terms are equal, so we can set up the following equations:
\(\[ a + 42 = ar + 32 \]\)
\(\[ ar + 32 = ar^2 + 2 \]\)
To solve these equations, let's begin by isolating \(\( a \)\) in terms of \(\( r \)\) from the first equation:
\(\[ a + 42 = ar + 32 \]\)
\(\[ a (1 - r) = -10 \]\)
\(\[ a = \frac{-10}{1 - r} \]\)
Next, substitute \(\( a \)\) into the second equation:
\(\[ ar + 32 = ar^2 + 2 \]\)
\(\[ \frac{-10r}{1 - r} + 32 = \frac{-10r^2}{1 - r} + 2 \]\)
We can simplify this equation by clearing the denominators. Let's multiply each term by \(\( 1 - r \)\):
\(\[ -10r + 32(1 - r) = -10r^2 + 2(1 - r) \]\)
\(\[ -10r + 32 - 32r = -10r^2 + 2 - 2r \]\)
\(\[ -10r + 32 - 32r = -10r^2 + 2 - 2r \]\)
\(\[ -42r + 32 = -10r^2 + 2 - 2r \]\)
\(\[ -42r + 32 = -10r^2 + 2 - 2r \]\)
\(\[ -42r + 32 = -10r^2 + 2 - 2r \]\)
\(\[ -42r + 32 = -10r^2 + 2 - 2r \]\)
Combining like terms:
\(\[ -42r + 32 = -10r^2 + 2 - 2r \]\)
\(\[ -40r + 32 = -10r^2 + 2 \]\)
\(\[ -40r = -10r^2 - 30 \]\)
\(\[ 4 =r \]\)
\(\[r = \frac{-30}{-(r -1)} =3\]\)
Therefore, the common ratio \(\( r = -3 \)\).
Now, substitute \(\( r \)\) back into the expression for \(\( a \)\):
\(\[ a = \frac{-10}{1 - (-3)} \]\)
\(\[ a = \frac{-10}{1 + 3} \]\)
\(\[ a = \frac{-10}{4} \]\)
\(\[ a = -2.5 \]\)
So, the three terms of the geometric sequence are:
- First term: \(\( a = -2.5 \)\)
- Second term: \(\( ar = -2.5 \times -3 = 7.5 \)\)
- Third term: \(\( ar^2 = 7.5 \times -3 = -22.5 \)\)
When we apply the modifications given in the problem:
- The first term modified is \(\( -2.5 + 42 = 39.5 \)\)
- The second term modified is \(\( 7.5 + 32 = 39.5 \)\)
- The third term modified is \(\( -22.5 + 2 = -20.5 \)\)
Thus, the values for the three terms, after applying the modifications, are:
\(\[ -2.5, 7.5, -22.5 \]\)
\(\[ 39.5, 39.5, -20.5 \]\)
