# If the seventh term of an A.P. is zero, then $\frac{{{t_{37}}}}{{{t_{13}}}}$ is ? tn denotes the nth term of AP.

12 views

closed

If the seventh term of an A.P. is zero, then $\frac{{{t_{37}}}}{{{t_{13}}}}$ is ?

tn denotes the nth term of AP.

1. 3
2. 5
3. 1
4. 2

by (54.5k points)
selected

Correct Answer - Option 2 : 5

Concept:

Let us consider sequence a1, a2, a3 …. an is an A.P.

• Common difference “d”= a2 – a1 = a3 – a2 = …. = an – an – 1
• nth term of the A.P. is given by an = a + (n – 1) d
• Sum of the first n terms = $S_n= \frac{n}{2}[2a+(n-1)\times d]$ = $\rm\frac{n}{2}[a+l]$

Where, a = First term, d = Common difference, n = number of terms, an = nth term and l = Last term

Calculation:

Let the first term of AP be 'a' and the common difference be 'd'

Given: Ninth term of an A.P. is zero

⇒ a7 = 0

⇒ a + (7 - 1) × d = 0

⇒ a + 6d = 0

∴ a = -6d           ----(1)

To find: $\frac{{{t_{37}}}}{{{t_{13}}}}$

⇒ $\frac{{{t_{37}}}}{{{t_{13}}}} = \frac{{a + \;36d}}{{a + 12d}}$

⇒ $\frac{{{t_{37}}}}{{{t_{13}}}} = \frac{{ - 6d\; + \;36d}}{{ - 6d\; + \;12d}}$

⇒ $\frac{{{t_{37}}}}{{{t_{13}}}} = \frac{{\;30d}}{{\;6d}}$

∴ $\frac{{{t_{37}}}}{{{t_{13}}}}$ = 5