Answer to Question #256955 in Calculus for junior

Question

1. From question b above, As a farmer, can you come out with a general term that can be used to forecast the height of the plant at any particular time? Explain mathematically.
2. A geometric progression has 6 terms. The first term is 192 and the common ratio is 1.5. An arithmetic progression has 21 terms and common difference 1.5. Given that the sum of all the terms in the geometric progression is equal to the sum of all the terms in the arithmetic progression, find the first term and the last term of the arithmetic progression.

Answer 1

Solution:

1. Question 1 is incomplete. It is related with some previous part. It cannot be answered as of now.
2. For Geometric Progression (G.P);

$a_1=192,n=6,r=1.5$

for Arithmetic Progression (A.P);

$d=1.5,n=21$

Also, sum of all terms in G.P = sum of all terms in A.P

Or $S_{n,G.P}=S_{n,A.P}$

$\Rightarrow \dfrac{a_1(r^n-1}{r-1}=\dfrac{n}{2}[2a_1+(n-1)d] \\ \Rightarrow \dfrac{192(1.5^6-1}{1.5-1}=\dfrac{21}{2}[2a_1+(21-1)1.5] \\ \Rightarrow 3990=\dfrac{21}{2}[2a_1+30] \\ \Rightarrow 2a_1+30=380 \\ \Rightarrow 2a_1=350 \\ \Rightarrow a=175$

Last term of A.P$=a_n=a+(n-1)d$

$=175+(21-1)1.5 \\=175+30 \\=205$