+1245
It took Lara five days to read a novel. Each day after the first day, Lara read half as many pages as the day before. If the novel was 248 pages long, how many pages did she read on the first day?
n + (1/2)n +(1/4)n + (1/8)n +(1/16)n=248, solve for n
n=128 pages Lara read on first day. So that you have:
128, 64, 32, 16, 8 =248
+26393 It took Lara five days to read a novel. Each day after the first day,
Lara read half as many pages as the day before.
If the novel was 248 pages long,
how many pages did she read on the first day?
Sequence is a geometric sequence:
Formula:
\(a_n = a\cdot r^{n-1} \\ sum~ s_n = \dfrac{a\left(1-r^{n+1}\right)} {1-r} \)
\(\begin{array}{|rcll|} \hline s_n &=& \dfrac{a\left(1-r^{n}\right)} {1-r} \quad & | \quad n=5,~ r=\dfrac12,~ s_n = 248 \\\\ 248 &=& \dfrac{a\left(1-\left(\dfrac12 \right)^{5}\right)} {1-\dfrac12} \\\\ 248 &=& \dfrac{a\left(1-\dfrac{1}{2^5}\right)} {\dfrac12} \\\\ 248 &=& 2a\left(1-\dfrac{1}{32}\right) \quad & | \quad :2 \\\\ 124 &=& a\left(1-\dfrac{1}{32}\right) \\\\ a\left(1-\dfrac{1}{32}\right) &=& 124 \\\\ a\left(\dfrac{32-1}{32}\right) &=& 124 \\\\ a\left(\dfrac{31}{32}\right) &=& 124 \quad & | \quad \cdot \dfrac{32}{31} \\\\ a &=& 124\cdot \dfrac{32}{31} \\\\ \mathbf{a} & \mathbf{=} & \mathbf{128} \\ \hline \end{array}\)
Lara read 128 pages on the first day.
