1. **Problem statement:** The first, seventh, and twenty-fifth terms of an arithmetic progression (A.P.) form the first three consecutive terms of a geometric progression (G.P.). The twentieth term of the A.P. is 22. We need to find the first term and common difference of the A.P. 2. **Formulas and definitions:** - The $n^{th}$ term of an A.P. is given by $$a_n = a + (n-1)d$$ where $a$ is the first term and $d$ is the common difference. - For three consecutive terms $x$, $y$, $z$ in a G.P., the relation $$y^2 = xz$$ holds. 3. **Express the given terms:** - First term: $a_1 = a$ - Seventh term: $a_7 = a + 6d$ - Twenty-fifth term: $a_{25} = a + 24d$ 4. **Apply the G.P. condition:** Since $a_1$, $a_7$, and $a_{25}$ are consecutive terms of a G.P., $$ (a_7)^2 = a_1 \times a_{25} $$ Substitute: $$ (a + 6d)^2 = a \times (a + 24d) $$ 5. **Expand and simplify:** $$ a^2 + 12ad + 36d^2 = a^2 + 24ad $$ Subtract $a^2$ from both sides: $$ 12ad + 36d^2 = 24ad $$ Bring all terms to one side: $$ 36d^2 = 24ad - 12ad = 12ad $$ Divide both sides by 12d (assuming $d \neq 0$): $$ 3d = a $$ 6. **Use the twentieth term condition:** The twentieth term is 22: $$ a_{20} = a + 19d = 22 $$ Substitute $a = 3d$: $$ 3d + 19d = 22 $$ $$ 22d = 22 $$ $$ d = 1 $$ 7. **Find the first term:** $$ a = 3d = 3 \times 1 = 3 $$ **Final answer:** - First term $a = 3$ - Common difference $d = 1$