SL 1.3—Geometric Sequences

The common ratio is found by dividing any term by the term immediately before it: $r=\frac{u_{n+1}}{u_n}$

Here $u_1=5$ and $u_2=4.85$, so: $r=\frac{4.85}{5}=0.97$

The common ratio is 0.97. Since $0<r<1$, the terms will decrease towards zero, which is consistent with going from 5 to 4.85.

We need $r=\frac{u_2}{u_1}=\frac{4.85}{5}$. The sequence is decreasing, so $r$ must be between 0 and 1.

If $r=0.03$, then $u_2=5\times 0.03=0.15\ne 4.85$. This is far too small. Eliminate 0.03.

If $r=1.03$, the sequence would be increasing: $u_2=5\times 1.03=5.15\ne 4.85$. Eliminate 1.03.

If $r=-0.97$, then $u_2=5\times (-0.97)=-4.85$, which is negative. The second term is positive. Eliminate -0.97.

The only remaining option is 0.97, and indeed $5\times 0.97=4.85$. ✓

The first term is $u_1=3$ and the common ratio is $r=\frac{12}{3}=4$.

$u_n=u_1\cdot r^{n-1}$ Substituting $n=7$: $u_7=3\times 4^{7-1}=3\times 4^6$

$4^6=4096$, so $u_7=3\times 4096=12288$.

The 7th term is 12288.

We have $u_1=3$, $r=4$, and we need $u_7=3\times 4^6$.

$768=3\times 256=3\times 4^4$, which corresponds to $u_5$, not $u_7$. Eliminate 768.

$3072=3\times 1024=3\times 4^5$, which corresponds to $u_6$. Eliminate 3072.

$9216=3\times 3072$. This does not fit $3\times 4^k$ for an integer $k$. Eliminate 9216.

$3\times 4^6=3\times 4096=12288$. ✓