Question

In an AP sixth term is one more than twice the third term
The Sum, of the fourth &fifth terms is five times the second term
find the tenth term of the Arithmetic Progression ?​

Answer 1

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

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

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.

According to statement,

↝ 6ᵗʰ term is 1 more than twice the third term.

$\rm :\longmapsto\:a_6 = 1 + 2a_3$

$\rm :\longmapsto\:a + (6 -1)d = 1 + 2(a + (3 - 1)d)$

$\rm :\longmapsto\:a +5d = 1 + 2(a + 2d)$

$\rm :\longmapsto\:a +5d = 1 + 2a + 4d$

$\bf\implies \:a = d - 1 - - - (1)$

According to statement,

Sum of fourth and fifth term is 5 times the second term.

$\rm :\longmapsto\:a_4 + a_5 = 2a_2$

$\rm :\longmapsto\:a + (4 - 1)d + a + (5 - 1)d = 5(a + (2 - 1)d)$

$\rm :\longmapsto\:a +3d + a + 4d = 5a + 5d$

$\rm :\longmapsto\:2a +7d = 5a + 5d$

$\rm :\longmapsto\:3a = 2d$

$\rm :\longmapsto\:3(d - 1)= 2d$

$\rm :\longmapsto\:3d - 3= 2d$

$\rm :\longmapsto\:3d - 2d= 3$

$\bf\implies \:d = 3$

On substituting the value of d, in equation (1), we get

$\bf\implies \:a = 3 - 1 = 2$

Hence,

↝ 10ᵗʰ term is,

$\rm :\longmapsto\:a_{10} = a + (10 - 1)d$

$\rm :\longmapsto\:a_{10} = a + 9d$

$\rm :\longmapsto\:a_{10} = 2 + 9 \times 3$

$\rm :\longmapsto\:a_{10} = 2 + 27$

$\rm :\longmapsto\:a_{10} = 29$

Additional Information :-

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of first n terms of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg(2\:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}$

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.

Answer 2

Answer:

Step-by-step explanation:

↝ nᵗʰ term of an arithmetic sequence is,

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.

According to statement,

↝ 6ᵗʰ term is 1 more than twice the third term.

According to statement,

Sum of fourth and fifth term is 5 times the second term.

On substituting the value of d, in equation (1), we get

Hence,

↝ 10ᵗʰ term is,

Additional Information :-

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of first n terms of an arithmetic sequence is,

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.