Verify that each of the following is an AP, and then write its next three terms.
$ 5, \frac{14}{3}, \frac{13}{3}, 4, \ldots $
Given:
Given sequence is \( 5, \frac{14}{3}, \frac{13}{3}, 4, \ldots \)
To do:
We have to verify whether the given sequence is an AP and write its next three terms.
Solution:
In the given sequence,
$a_1=5, a_2= \frac{14}{3}, a_3=\frac{13}{3}, a_4=4$
$a_2-a_1=\frac{14}{3}-5=\frac{14-3(5)}{3}=\frac{14-15}{3}=\frac{-1}{3}$
$a_3-a_2=\frac{13}{3}-\frac{14}{3}=\frac{13-14}{3}=\frac{-1}{3}$
$a_4-a_3=4-\frac{14}{3}=\frac{4(3)-13}{3}=\frac{12-13}{3}=\frac{-1}{3}$
Therefore,
$a_2-a_1=a_3-a_2=a_4-a_3$
The given sequence is an AP.
$d=\frac{-1}{3}$
$a_5=a_4+d=4+\frac{-1}{3}=\frac{4(3)-1}{3}=\frac{12-1}{3}=\frac{11}{3}$
$a_6=a_5+d=\frac{11}{3}+\frac{-1}{3}=\frac{11-1}{3}=\frac{10}{3}$
$a_7=a_6+d=\frac{10}{3}+\frac{-1}{3}=\frac{10-1}{3}=\frac{9}{3}=3$
The next three terms of the given sequence are $\frac{11}{3}, \frac{10}{3}$ and $3$.
Related Articles
- Verify that each of the following is an AP, and then write its next three terms.\( 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, \ldots \)
- Verify that each of the following is an \( \mathrm{AP} \), and then write its next three terms.\( \sqrt{3}, 2 \sqrt{3}, 3 \sqrt{3}, \ldots \)
- Verify that each of the following is an \( \mathrm{AP} \), and then write its next three terms.\( a, 2 a+1,3 a+2,4 a+3, \ldots \)
- Verify that each of the following is an \( \mathrm{AP} \), and then write its next three terms.\( a+b,(a+1)+b,(a+1)+(b+1), \ldots \)
- Fill in the blanks:(i) \( \frac{-4}{13}-\frac{-3}{26}= \)(ii) \( \frac{-9}{14}+\ldots \ldots \ldots . . .=-1 \)(iii) \( \frac{-7}{9}+\ldots \ldots . . . .=3 \)(iv) ............ \( +\frac{15}{23}=4 \)
- Find the sum of the two middle most terms of the AP: \( -\frac{4}{3},-1,-\frac{2}{3}, \ldots .4 \frac{1}{3} \).
- Solve each of the following equations and also verify your solution:(i) $\frac{(2x-1)}{3}-\frac{(6x-2)}{5}=\frac{1}{3}$(ii) $13(y-4)-3(y-9)-5(y+4)=0$
- Which of the following form an AP? Justify your answer.\( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots \)
- Find the common difference and write the next four terms of each of the following arithmetic progressions :$-1, \frac{1}{4}, \frac{3}{2}, ……..$
- Which is greater in each of the following:$(i)$. $\frac{2}{3},\ \frac{5}{2}$$(ii)$. $-\frac{5}{6},\ -\frac{4}{3}$$(iii)$. $-\frac{3}{4},\ \frac{2}{-3}$$(iv)$. $-\frac{1}{4},\ \frac{1}{4}$$(v)$. $-3\frac{2}{7\ },\ -3\frac{4}{5}$
- Find the common difference and write the next four terms of each of the following arithmetic progressions :$-1, -\frac{5}{6}, -\frac{2}{3}, ………..$
- For the following APs, write the first term and the common difference:$\frac{1}{3}, \frac{5}{3}, \frac{9}{3}, \frac{13}{3}, ……..$
- Evaluate each of the following:(i) \( \frac{2}{3}-\frac{3}{5} \)(ii) \( \frac{-4}{7}-\frac{2}{-3} \)(iii) \( \frac{4}{7}-\frac{-5}{-7} \)(iv) \( -2-\frac{5}{9} \)(v) \( \frac{-3}{-8}-\frac{-2}{7} \)(vi) \( \frac{-4}{13}-\frac{-5}{26} \)(vii) \( \frac{-5}{14}-\frac{-2}{7} \).(viii) \( \frac{13}{15}-\frac{12}{25} \)(ix) \( \frac{-6}{13}-\frac{-7}{13} \)(x) \( \frac{7}{24}-\frac{19}{36} \)(xi) \( \frac{5}{63}-\frac{-8}{21} \)
- If $cos\ A = \frac{4}{5}$, then the value of $tan\ A$ is(A) $\frac{3}{5}$(B) $\frac{3}{4}$(C) $\frac{4}{3}$(D) $\frac{5}{3}$
- Verify the Following by using suitable property: $(\frac{5}{4}+\frac{-1}{2})+\frac{-3}{2}= \frac{5}{4}+(\frac{-1}{2}+\frac{-3}{2})$
Kickstart Your Career
Get certified by completing the course
Get Started