Question

The 5th term of an A.P. exceeds twice the 2nd term by 1. The 10th term exceeds twice the 4

th term by 3. Find the first term and the common of the A.P.​

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.

Tʜᴜs,

Let assume that,

• First term of an AP is a

• Common Difference of an AP is d.

According to statement,

The 5th term of an A.P. exceeds twice the 2nd term by 1.

$\rm :\longmapsto\:a_5 - 2a_2 = 1$

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

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

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

$\rm :\longmapsto\: - a + 2d = 1$

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

Also,

According to statement,

The 10th term exceeds twice the 4th term by 3

$\rm :\longmapsto\:a_{10} - 2a_4 = 3$

$\rm :\longmapsto\:a + (10 - 1)d - 2\bigg(a + (4 - 1)d\bigg) = 3$

$\rm :\longmapsto\:a + 9d - 2\bigg(a + 3d\bigg) = 3$

$\rm :\longmapsto\:a + 9d - 2a - 6d = 3$

$\rm :\longmapsto\: - a + 3d = 3$

$\rm :\longmapsto\: - (2d - 1) + 3d = 3 \: \: \: \: \: \{ \: using \: (1) \: \}$

$\rm :\longmapsto\: - 2d + 1 + 3d = 3$

$\rm :\longmapsto\: d = 3 - 1$

$\bf :\longmapsto\: d = 2$

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

$\rm :\longmapsto\: a = 2(2) - 1$

$\rm :\longmapsto\: a = 4 - 1$

$\bf :\longmapsto\: a = 3$

Hence,

First term of an AP is 3

and

Common Difference of an AP is 2.

Additional Information :-

↝ Sum of 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 n terms of AP.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

Answer 2

Answer:

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic sequence is,

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

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.

Tʜᴜs,

Let assume that,

First term of an AP is a

Common Difference of an AP is d.

According to statement,

The 5th term of an A.P. exceeds twice the 2nd term by 1.

\rm :\longmapsto\:a_5 - 2a_2 = 1:⟼a5−2a2=1

\rm :\longmapsto\:a + (5 - 1)d - 2\bigg(a + (2 - 1)d\bigg) = 1:⟼a+(5−1)d−2(a+(2−1)d)=1

\rm :\longmapsto\:a + 4d - 2\bigg(a + d\bigg) = 1:⟼a+4d−2(a+d)=1

\rm :\longmapsto\:a + 4d - 2a - 2 d = 1:⟼a+4d−2a−2d=1

\rm :\longmapsto\: - a + 2d = 1:⟼−a+2d=1

\rm :\longmapsto\: a = 2d - 1 - - - (1):⟼a=2d−1−−−(1)

Also,

According to statement,

The 10th term exceeds twice the 4th term by 3

\rm :\longmapsto\:a_{10} - 2a_4 = 3:⟼a10−2a4=3

\rm :\longmapsto\:a + (10 - 1)d - 2\bigg(a + (4 - 1)d\bigg) = 3:⟼a+(10−1)d−2(a+(4−1)d)=3

\rm :\longmapsto\:a + 9d - 2\bigg(a + 3d\bigg) = 3:⟼a+9d−2(a+3d)=3

\rm :\longmapsto\:a + 9d - 2a - 6d = 3:⟼a+9d−2a−6d=3

\rm :\longmapsto\: - a + 3d = 3:⟼−a+3d=3

\rm :\longmapsto\: - (2d - 1) + 3d = 3 \: \: \: \: \: \{ \: using \: (1) \: \}:⟼−(2d−1)+3d=3{using(1)}

\rm :\longmapsto\: - 2d + 1 + 3d = 3:⟼−2d+1+3d=3

\rm :\longmapsto\: d = 3 - 1:⟼d=3−1

\bf :\longmapsto\: d = 2:⟼d=2

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

\rm :\longmapsto\: a = 2(2) - 1:⟼a=2(2)−1

\rm :\longmapsto\: a = 4 - 1:⟼a=4−1

\bf :\longmapsto\: a = 3:⟼a=3

Hence,

First term of an AP is 3

and

Common Difference of an AP is 2.

Additional Information :-

↝ Sum of n  terms of an arithmetic sequence is,

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

Wʜᴇʀᴇ,

Sₙ is the sum of n terms of AP.

a is the first term of the sequence.

n is the no. of terms.

d is the common difference

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