Question

The average of 10 terms and
12 terms of an AP is 12 and
14 respectively. 5^th term of
the progression will be?​

Answer 1

Answer:

answer for 5th term will be 5 as the average reveals this fact soon

Answer 2

Answer:

The 5th term of the AP = 11

Step-by-step explanation:

Given,

The average of 10 terms and 12 terms of an AP is 12 and 14 respectively

To find,

The fifth term of the AP

Recall the formula

Average = $\frac{Sum \ of \ observations}{Total \ no \ of \ observations }$

Sum of n terms of an AP = $\frac{n}{2} [2a+(n-1)d]$

nth term of an AP = a+(n-1)d, where 'a' is the first term and 'd' is the common difference

Solution:

Average of first 10 terms of an AP = $\frac{Sum \ of \ 10 \ terms}{10}$

Sum of 10 terms = $\frac{10}{2}[{2a+(10-1)d]$ = 5[2a +9d]

Average of first 10 terms  = $\frac{5[2a +9d]}{10}$ = $\frac{1}{2}[2a+9d]$

Since The average of 10 terms is 12, we have  $\frac{1}{2}[2a+9d]$ = 12

2a+9d = 24 --------------------(1)

Average of first 12 terms of an AP = $\frac{Sum \ of \ 12 \ terms}{12}$

Sum of 12 terms = $\frac{12}{2}[{2a+(12-1)d]$ = 6[2a +11d]

Average of first 12 terms  = $\frac{6[2a +11d]}{12}$ = $\frac{1}{2}[2a+11d]$

Since The average of 12 terms is 14, we have  $\frac{1}{2}[2a+11d]$ = 14

2a+11d = 28 --------------------(2)

(2) - (1) → 2d = 4

d = 2

(1)  → 2a+9×2 = 24

2a = 24 - 18 = 6

a = 3

First term of the AP = 3 and common difference = 2

5th term of the AP = a+4d = 3+4×2 = 3+8 = 11

∴The 5th term of the AP = 11

#SPJ3