Question

In the series $357,363,369,....,$what will be the $10th$term?
A.$405$
B.$411$
C.$413$
D.$417$

Answer 1

Hint: In arithmetic progression the difference between the two consecutive terms is also constant whereas in the geometric progression the ratio of two consecutive terms remains the constant. First of all identify the given sequence and then use the standard formula for the required tenth term for the series.

Complete step-by-step answer:
Take the given series: $357,363,369,....,$
Now, find the difference between two consecutive terms in the series –
$
  d = 363 - 357 = 6 \\
  d = 369 - 363 = 6 \;
 $
Since the difference between the terms remains constant, the given series is in arithmetic progression.
Nth term in the arithmetic progression can be given by –
${a_n} = a + (n - 1)d$
Here, first term $a = 357$
Difference, $d = 6$
$n = 10$
So, tenth term is –
${a_{10}} = 357 + (10 - 1)6$
Simplify the above expression finding the difference of the terms –
${a_{10}} = 357 + (9)6$
Find the product of the terms in the above expression –
${a_{10}} = 357 + 54$
Simplify finding the sum of the terms –
\[{a_{10}} = 411\]
From the given multiple choices – the option B is the correct answer.
So, the correct answer is “Option B”.

Note: Be careful while placing the values in the standard formula and remember the standard formula properly. Remember the multiples of the numbers at least till twenty and also be careful while simplifying. The nth term of the geometric progression can be given by ${a_n} = a{r^{n - 1}}$