Question

What is the sum of the $12th$ to $20th$ terms (inclusive) of the arithmetic sequence $7,12,17,22,...?$
$A)516$
$B)676$
$C)117$
$D)667$

Answer 1

Hint: While talking about the A.P and G.P, we need to know about the concept of Arithmetic and Geometric progression.
An arithmetic progression can be represented by $a,(a + d),(a + 2d),(a + 3d),...$where $a$ is the first term and $d$ is a common difference.
A geometric progression can be given by $a,ar,a{r^2},....$ where $a$ is the first term and $r$ is a common ratio.
Here in this given question, we clearly see that the difference is $5$ and they give arithmetic progress.
Formula used: The sum of the n-terms of the A.P is ${S_n} = (\dfrac{n}{2})(2a + (n - 1)d)$

Complete step-by-step solution:
Given that we have the sequence as $7,12,17,22,...$ and which is clearly as AP.
To find the first number of the sequence is very easy. Clearly, we have the first term as $a = 7$ and then to find the common difference we will subtract the second value and first value then we have $d = 12 - 7 = 5$ and hence the common difference is $d = 5$
Now to find the sum of the $12th$ term of the given sequence we use the formula ${S_n} = (\dfrac{n}{2})(2a + (n - 1)d)$
Put $n = 12$ then we have ${S_{12}} = (\dfrac{{12}}{2})(2a + (12 - 1)d)$ and also, we know that $a = 7,d = 5$ then we get ${S_{12}} = (\dfrac{{12}}{2})(2(7) + (12 - 1)(5))$
Now simplifying the equation, we have ${S_{12}} = 6(14 + (11)(5)) = 6(14 + 55)$
${S_{12}} = 6(69)= 414$
Now to find the sum of the $20th$ term of the given sequence we use the formula ${S_n} = (\dfrac{n}{2})(2a + (n - 1)d)$
Put $n = 20$ then we have ${S_{20}} = (\dfrac{{20}}{2})(2a + (20 - 1)d)$ and also, we know that $a = 7,d = 5$ then we get \[{S_{20}} = (\dfrac{{20}}{2})(2(7) + (20 - 1)(5))\]
Now simplifying the equation, we have ${S_{20}} = 10(14 + (19)(5)) = 10(14 + 95)$
${S_{20}} = 10(109)= 1090$
Hence, we have the sum of $12$ term as ${S_{12}} = 414$ and sum of $20$ term as ${S_{20}} = 1090$
Therefore, the sum of these two terms from $12th$ to $20th$ terms (inclusive), which is $12,13,14,...,20$ means the difference of the values. Hence, we have to find the difference using the subtraction of the largest value minus the smallest values.
Hence, we have the sum of the $12th$ to $20th$ terms as $1090 - 414 = 676$
Therefore, the option $B)676$ is correct.

Note: Similarly, the Geometric Progression:
$\bullet$ In the GP the series is obtained by multiplying two consecutive terms so that they have constant factors.
$\bullet$ In GP the series is identified with the help of a common ratio between consecutive terms.
$\bullet$ Series vary in the exponential form because it increases by multiplying the terms.
For GP with the common ratio the formula to be calculated $GP = \dfrac{a}{{r - 1}},r \ne 1,r < 0$and $GP = \dfrac{a}{{1 - r}},r \ne 1,r > 0$
Harmonic progress is the reciprocal of the given arithmetic progression which is the form of $HP = \dfrac{1}{{[a + (n - 1)d]}}$where $a$ is the first term and $d$ is a common difference and n is the number of AP.