Question

if the general term of an ap is 9 - 5n and find the sum of first 10 terms of AP​

Answer 1

Answer:Given: a

n

​

=9−5n

Taking n=1,

a

1

​

=9−5(1)=9−5=4

Taking n=2,

a

2

​

=9−5(2)=9−10=−1

Taking n=3,

a

3

​

=9−5(3)=9−15=−6

Therefore the series is 4,−1,−6,...

So,a=4,d=a

2

​

−a

1

​

=−1−4=−5

Now, we have to find the sum of the first 15

th

terms of the AP

S

n

​

=

2

n

​

[2a+(n−1)d]

⇒S

n

​

=

2

15

​

[2×4+(15−1)(−5)]

⇒S

15

​

=

2

15

​

[8−70]

⇒S

15

​

=

2

15

​

[−62]

⇒S

15

​

=15×(−31)

⇒S

15

​

=−465

Hence, the sum of 15

th

terms is −465.

Step-by-step explanation:

Answer 2

$\large\underline{\sf{Given- }}$

$\rm :\longmapsto\:An \: AP \: series \: whose \: a_n = 9 - 5n$

$\large\underline{\sf{To\:Find - }}$

$\rm :\longmapsto\:Sum \: of \: 10 \: terms, \: S_{10}$

$\begin{gathered}\Large{\sf{{\underline{Formula \: Used - }}}}  \end{gathered}$

Sum of n terms of an AP series is given by

$\rm :\longmapsto\: S_{n} = \dfrac{n}{2} \bigg(2a + (n - 1)d \bigg)$

Wʜᴇʀᴇ,

• Sₙ is the Sum of first 'n' terms.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

$\large\underline{\sf{Solution-}}$

Given that,

$\rm :\longmapsto\:a_n = 9 - 5n$

Let first find an AP series,

For n = 1

$\rm :\longmapsto\:a_1 = 9 - 5 \times 1$

$\rm :\longmapsto\:a_1 = 9 - 5$

$\bf\implies \:a_1 = 4$

For n = 2

$\rm :\longmapsto\:a_2 = 9 - 5 \times 2$

$\rm :\longmapsto\:a_2 = 9 - 10$

$\rm :\longmapsto\:a_2 = - 1$

For n = 3

$\rm :\longmapsto\:a_3 = 9 - 5 \times 3$

$\rm :\longmapsto\:a_3 = 9 - 15$

$\rm :\longmapsto\:a_3 = - 6$

• So, AP series is

$\rm :\longmapsto\:4, \: - 1, \: - 6, - - -$

So, we have

• First term of an AP, a = 4

• Common Difference of an AP, d = - 1 - 4 = - 5

So,

• Sum of 'n' terms is given by

$\rm :\longmapsto\: S_{n} = \dfrac{n}{2} \bigg(2a + (n - 1)d \bigg)$

Now, Substitute the values

• a = 4

• d = - 5

• n = 10

Thus,

$\rm :\longmapsto\: S_{10} = \dfrac{10}{2} \bigg(2 \times 4 + (10 - 1) \times ( - 5) \bigg)$

$\rm :\longmapsto\: S_{10} = 5 \times (8 - 45)$

$\rm :\longmapsto\: S_{10} = 5 \times ( - 37) = - 185$

Additional Information :-

↝ nᵗʰ term of an arithmetic sequence is,

$\rm :\longmapsto\:\begin{gathered}\:\:{\underline{{\boxed{\bf{{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.