This problem will seriously make you feel stupid. Just a warning.
Since it wouldn’t fit in one image:
Let bn be a geometric series such that $b_1+b_{10}=9$ and $b_2 + b_3 + \cdots + b_9 = 10$. Find the value of: $$ \frac{b_1^2+b_1b_2+10b_2}{b_1} $$
Really nothing to say here, besides solving for a or r in the geometric series is unnecessary (since we know that a geometric series follows the form $\sum_{n\geq1} ar^{n-1}$).
The key to solving problems like this is to try and relate parts of the missing value to the statements given to us. We can then try and rearrange or piece together these givens into the expression we want to find to see if anything comes up. So yes, it kind of boils down to RNG to put it simply.
Many of these problems that have a very simple wording and a very simple expression to find are what I like to call “tipping point problems”, since there’s always one clue that you struggle so much to find, but once the clue is apparent, everything else falls right into place.
At first glance, the fraction seems completely arbitrary and seems to follow no pattern at all.
However, if we look at the fraction, we can try relating the 10 as the coefficient of the b2 to the equation b2 + b3 + … + b9 = 10. For me, the tipping point of this problem (which took far too long to figure out) was factoring out a b2 in the numerator like so: $$ \frac{b_1^2+b_2(b_1+10)}{b_1} $$
We can now see that if we add b1 to both sides of b2 + b3 + … + b9 = 10, we can obtain: $$ \frac{b_1^2+b_2(b_1+b_2+\cdots+b_9)}{b_1}=b_1+\frac{b_2}{b_1}(b_1+b_2+\cdots+b_9) $$
Finally, we can recognize that $\frac{b_2}{b_1}$ is just the ratio r of the geometric series. If we multiply a term bn by the series ratio r, we will just get bn+1. Therefore, we can apply that to the second term here and get that this whole expression is just equal to: $$ b_1+(b_{1+1}+b_{2+1}+\cdots + b_{9+1})=b_1+(b_2+b_3+\cdots+b_{10}) $$
Adding the two equations from the start, we obtain that $$ b_1+b_2+b_3+\cdots+b_{10}=\boxed{19} $$