Question

5) The mth term of an arithmetic
sequence is n and the nth term is m.
What is the common difference of this
sequence?
Ans: m = a + (n-1)d
n = a + (m-1)d
m-n-and-d-(a + md-d)
- nd-d-a-md + d = nd-md
=d(n-m)
mn mn
da
- 1
nem
Common difference --

can anybody explain how this answer came?​

Answer 1

✬ Common Difference = –1 ✬

Step-by-step explanation:

Given:

• mth term of an arithmetic sequence is n.
• nth term of an arithmetic sequence is m.

To Find:

• What is the common difference ?

Solution: As we know that the nth term an arithmetic sequence is given by

★ a + (n – 1)d ★

• a = First term

• n = number of terms

• d = difference between two terms

Now , a/q

➼ mth term is n

• am = a + (m – 1)d = nㅤㅤㅤㅤ∼eqⁿ(i)

➼ nth term is m

• aⁿ = a + (n – 1)d = mㅤㅤㅤㅤ∼eqⁿ(ii)

[ Subtracting the both equations ]

ㅤㅤㅤㅤa + (m – 1)d = n

ㅤㅤㅤㅤa + (n – 1)d = m

ㅤㅤㅤㅤ-ㅤ-ㅤㅤㅤㅤ-

ㅤㅤㅤ════════════

➨ (m – 1)d – (n – 1)d = (n – m)

➨ md – d – (nd – d) = (n – m)

➨ md – d – nd + d = (n – m)

➨ md – nd = (n – m)

➨ (m – n)d = (n – m)

➨ d = (n – m)/(m – n)

➨ d = (n – m)/–(n – m)

• Here after dividing 1 left above in numerator and – left below in denominator.

• As we know that – sign cannot be written in denominator so it will attach with numerator.

• So here we got the common difference as –1.

➨ d = –1

Hence, common difference of the arithmetic sequence is –1.

Answer 2

Gɪᴠᴇɴ :

• The mᵗʰ term of an arithmetic sequence is n.

• The nᵗʰ term of an arithmetic sequence is m.

Tᴏ Fɪɴᴅ :

• The common difference (d) of the given sequence.

Cᴀʟᴄᴜʟᴀᴛɪᴏɴ :

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic sequence is,

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

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,

↝ mᵗʰ term is,

➛ $\bf{a_m\:=\:a\:+\:(m\:-\:1)\:d}$

➣ It is given that mᵗʰ term is n.

➛ $\bf{\boxed{n\:=\:a\:+\:(m\:-\:1)\:d}}--(i)$

Aɢᴀɪɴ,

↝ nᵗʰ term is,

➙ $\bf{a_n\:=\:a\:+\:(n\:-\:1)\:d}$

➣ It is given that nᵗʰ term is m.

➙ $\bf{\boxed{m\:=\:a\:+\:(n\:-\:1)\:d}}--(ii)$

Nᴏᴡ,

↝ Subtracting equation (ii) from (i), we get

$:\rightarrow\:\bf{\Big[a\:+\:(m\:-\:1)\:d\Big]\:-\:\Big[a\:+\:(n\:-\:1)\:d\Big]\: = \:n\:-\:m\:}$

$:\rightarrow\:\bf{a\:+\:(m\:-\:1)\:d\:-\:a\:-\:(n\:-\:1)\:d\: = \:n\:-\:m\:}$

$:\rightarrow\:\bf{a\:-\:a\:+\:(m\:-\:1)\:d\:-\:(n\:-\:1)\:d\: = \:n\:-\:m\:}$

$:\rightarrow\:\bf{(m\:-\:1)\:d\:-\:(n\:-\:1)\:d\: = \:n\:-\:m\:}$

$:\rightarrow\:\bf{md\:-\:d\:-\:nd\:+\:d\: = \:n\:-\:m\:}$

$:\rightarrow\:\bf{(m\:-\:n)\:d\: = \:n\:-\:m\:}$

$:\rightarrow\:\bf{(m\:-\:n)\:d\: = \:-\:(m\:-\:n)\:}$

$:\rightarrow\:\bf{d\: = \:\dfrac{-\:(m\:-\:n)}{(m\:-\:n)}\:}$

$:\rightarrow\:\bf\pink{d\: = \:-\:1\:}$

$\Large\bf{Therefore,}$

The common difference of the given sequence is -1.