4. The numbers $a, b, c, d$ are in arithmetic progression, while the numbers $a+1, b-1, c-1, d+3$ are in geometric progression. Find the numbers.

Grade 9 Manual

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## Grading Rubric OML Grade 9, 2014
4. Let $r=$ the common difference of the arithmetic progression, $q=$ the common ratio of the geometric progression $\Rightarrow b=a+r$,

$c=a+2 r, d=a+3 r$

$b-1=(a+1) q, c-1=(a+1) q^{2}, d+3=(a+1) q^{3} \Rightarrow\left\{\begin{array}{l}a+r-1=(a+1) q \\ a+2 r-1=(a+1) q^{2} \ldots .1 p \\ a+3 r+3=(a+1) q^{3}\end{array}\right.$

From equations 1 and $2 \Rightarrow \mathrm{r}=\mathrm{q}(\mathrm{q}-1)(\mathrm{a}+1)$ $1 p$

From equations 2 and $3 \Rightarrow r+4=(a+1) q^{2}(q-1)=r q \Rightarrow q=\frac{r+4}{r}$

Substitute $\mathrm{r}, \mathrm{q}$ into the system $\Rightarrow \mathrm{r}=4$ $.1 \mathrm{p}$

$\Rightarrow \mathrm{q}=2 \Rightarrow \mathrm{a}=1$.................................................................................................. $1 \mathrm{p}$

The numbers are $\mathrm{a}=1, \mathrm{~b}=5, \mathrm{c}=9, \mathrm{~d}=13$