I can't understand one thing in one question of Arithmetic Progression. I have to find $21^{st}$ term from the set $\{12,2,-4,-10\}$ and this is where problem start.
Arithmetic Progression
Equation: $T_n=a+(n-1)d$
$Tn$ = Term
$n$ = Term Number
$a$ = First Number
$d$ = Difference between two series wise numbers
I need this data to find out mentioned number, like $21^{st}$ number, if I find just $d$ I can just count the numbers to see what number is coming at $21^{st}$ position, but in this set of numbers, has unmatched difference and I don't know why, either it is right set or wrong.
Let's take example;
If I want to find $5^{th}$ term from set $\{2,4,6...\}$ then I need all the data listed above and if I find just $d$ from this set, I can figure out $5^{th}$ number without using equation.
Here is example;
$d=b-a$ | a is first term and b is second term
$d=4-2$
$d=2$
so the difference between two "series wise" numbers is $2$. Now in very first set "mentioned above" is different difference in first two terms and in second two terms.
$\{12,2,-4,-8\}$
so first two terms $\{12,2\}$ has difference of $10$
and next two terms $\{2,-4\}$ has difference of $6$.
Answer
To put this plainly, your sequence is not an arithmetic progression. If you set
$$
\{T_1,T_2,T_3,T_4\}=\{12,2,-4,-10\},
$$
then $T_2-T_1=-10$ and $T_3-T_2=-6$, which is completely inconsistent with an arithmetic progression. You also have $T_4-T_3=-6$, so you can sort of hope that the first $12$ is a typo (instead of an $8$), in which case the sequence goes as
$$T_n=12-6(n-1)$$
and the 21st term is then $T_{21}=-108$, but extrapolating based on four not-very-consistent terms is always a very dicy proposition. You will need to weigh this yourself with respect to the source you got this question from: how they phrase the section, how strongly they imply the progression is arithmetic, and how likely they are to have made a typo.
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