Question

Which step in the arithmetic sequences = \[\{ 4,7,10,13,{a_5},{a_{6,..}}\} \] involves 12% increase over the immediately preceding term?
a) \[{a_5}\] to \[{a_6}\]
b) \[{a_6}\] to \[{a_7}\]
c) \[{a_7}\] to \[{a_8}\]
d) \[{a_8}\] to \[{a_9}\]
e) \[{a_9}\] to \[{a_{10}}\]

Answer 1

Hint: Here the question is related to the arithmetic sequences. The sequence is already given in the question; the first term will be the \[{a_1}\] and the difference between the two successive terms will be the common difference d. Then by using the formula \[{a_n} = {a_1} + (n - 1)d\], we find the other terms of the sequence. Then by using the condition \[{a_n} = {a_{n - 1}} + \dfrac{{12}}{{100}} \times {a_{n - 1}}\] , we are going to determine the in which step the sequence involves 12% increase over the preceding term.

Complete step-by-step answer:
 In the sequence we have three kinds of sequence namely, arithmetic sequence, geometric sequence and harmonic sequence.
In arithmetic sequence we have the common difference between the two terms, In geometric sequence we have the common ratio between the two terms, In harmonic sequence it is a ratio of arithmetic sequence to geometric sequence.
The general arithmetic progression is of the form \[a,a + d,a + 2d,...\] where a is first term nth d is the common difference. The nth term of the arithmetic progression is defined as \[{a_n} = {a_1} + (n - 1)d\]
Now we consider the given sequence \[\{ 4,7,10,13,{a_5},{a_{6,..}}\} \]. First we determine the difference.
The difference between \[{a_1}\]and \[{a_2}\], so we have
\[d = {a_2} - {a_1} = 7 - 4 = 3\]
Therefore \[d = 3\]
We determine the values of \[{a_5}\], \[{a_6}\],\[{a_7}\], \[{a_8}\],\[{a_9}\] and \[{a_{10}}\]. Then we check the condition \[{a_n} = {a_{n - 1}} + \dfrac{{12}}{{100}} \times {a_{n - 1}}\] and this should to satisfied. Then we are choosing the option.
Now we consider the formula \[{a_n} = {a_1} + (n - 1)d\] and hence determine \[{a_5}\], \[{a_6}\],\[{a_7}\], \[{a_8}\],\[{a_9}\] and \[{a_{10}}\].
The values of \[{a_1} = 4\] and \[d = 3\]
Now we determine the value of \[{a_5}\]
\[ \Rightarrow {a_5} = 4 + (5 - 1)3\]
On simplifying we have
\[ \Rightarrow {a_5} = 16\]
Now we determine the value of \[{a_6}\]
\[ \Rightarrow {a_6} = 4 + (6 - 1)3\]
On simplifying we have
\[ \Rightarrow {a_6} = 19\]
Now we check the condition, here the value of n is 6.
\[ \Rightarrow {a_6} = {a_5} + \dfrac{{12}}{{100}} \times {a_5}\]
On substituting the values and simplifying we have
\[ \Rightarrow 19 = 16 + \dfrac{{12}}{{100}} \times 16\]
\[ \Rightarrow 19 \ne 17.92\]
Now we determine the value of \[{a_7}\]
\[ \Rightarrow {a_7} = 4 + (7 - 1)3\]
On simplifying we have
\[ \Rightarrow {a_7} = 22\]
Now we check the condition, here the value of n is 7.
\[ \Rightarrow {a_7} = {a_6} + \dfrac{{12}}{{100}} \times {a_6}\]
On substituting the values and simplifying we have
\[ \Rightarrow 21 = 19 + \dfrac{{12}}{{100}} \times 19\]
\[ \Rightarrow 21 \ne 21.28\]
Now we determine the value of \[{a_8}\]
\[ \Rightarrow {a_8} = 4 + (8 - 1)3\]
On simplifying we have
\[ \Rightarrow {a_8} = 25\]
Now we check the condition, here the value of n is 8.
\[ \Rightarrow {a_8} = {a_7} + \dfrac{{12}}{{100}} \times {a_7}\]
On substituting the values and simplifying we have
\[ \Rightarrow 25 = 21 + \dfrac{{12}}{{100}} \times 21\]
\[ \Rightarrow 25 \ne 23.52\]
Now we determine the value of \[{a_9}\]
\[ \Rightarrow {a_9} = 4 + (9 - 1)3\]
On simplifying we have
\[ \Rightarrow {a_8} = 28\]
Now we check the condition, here the value of n is 9.
\[ \Rightarrow {a_9} = {a_8} + \dfrac{{12}}{{100}} \times {a_8}\]
On substituting the values and simplifying we have
\[ \Rightarrow 28 = 25 + \dfrac{{12}}{{100}} \times 25\]
\[ \Rightarrow 28 = 28\]
The given condition is satisfied
Therefore \[{a_8}\] to \[{a_9}\] involves a 12% increase over the immediately preceding term.
Hence the option d) is the correct one.
So, the correct answer is “Option D”.

Note: The given condition should be written in the form of numerical. By checking each term will give the correct picture. Because the common difference is the same for every term but the percentage of increasing or decreasing rate will depend on the terms of the given sequence.