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