If (m+1)th term of an A.P. is twice the (n + 1)th term, prove that (3m + 1)th term is twice the (m + n + 1)th term.

Open in App

Solution

Given:
$a_{m + 1} = 2 a_{n + 1} \Rightarrow a + (m + 1 - 1) d = 2 [a + (n + 1 - 1) d] \Rightarrow a + m d = 2 (a + n d)$
$\Rightarrow a + m d = 2 a + 2 n d$
$0 = a + 2 n d - m d$
$\Rightarrow n d = \frac{m d - a}{2} \dots \dots (i)$
To prove:
$a_{3 m + 1} = 2 a_{m + n + 1}$
$L H S : a_{3 m + 1} = a + (3 m + 1 - 1) d$
$\Rightarrow a_{3 m + 1} = a + 3 m d$
$R H S : 2 a_{m + n + 1} = 2 [a + (m + n + 1 - 1) d]$
$\Rightarrow 2 a_{m + n + 1} = 2 (a + m d + n d) \Rightarrow 2 a_{m + n + 1} = 2 [a + m d + (\frac{m d - a}{2})]$
[From (i)]
$\Rightarrow 2 a_{m + n + 1} = 2 [\frac{2 a + 2 m d + m d - a}{2}] \Rightarrow 2 a_{m + n + 1} = 2 [\frac{a + 3 m d}{2}] \Rightarrow 2 a_{m + n + 1} = a + 3 m d$
$∴$ LHS = RHS

Suggest Corrections

Similar questions

If (m+1)th term of an A.P. is twice the (n+1)th term, prove that (3m+1)th term is twice the (m+n+1)th term.

{(m + 1)}^{t h}

{(n + 1)}^{t h}

{(3 m + 1)}^{t h}

{(m + n + 1)}^{t h}

(m + 1)

(n + 1)

72

34

(p + 1) t h

(q + 1) t h

(3 p + 1) t h

= m (p + 1) t h

m

Join BYJU'S Learning Program