Question 31 of 51intermediate🔧 ApplyNumerical3 marks

Find the 12th12^{th} term of a GP with common ratio 2, whose 8th8^{th} term is 192.

Correct Answer

Given: The common ratio of the GP is r=2r = 2 and the 8th term is t8=192t_8 = 192. The formula for the nnth term of a GP is tn=arn−1t_n = ar^{n-1}, where aa is the first term.

Step 1: Find the first term (aa) using the 8th term. Substituting n=8n = 8, r=2r = 2, and t8=192t_8 = 192 into the formula: 192=a(2)8−1⇒192=a(2)7⇒192=128a⇒a=192128=1.5192 = a(2)^{8-1} \Rightarrow 192 = a(2)^7 \Rightarrow 192 = 128a \Rightarrow a = \frac{192}{128} = 1.5.

Step 2: Calculate the 12th term (t12t_{12}). Using the formula tn=arn−1t_n = ar^{n-1} with a=1.5a = 1.5, r=2r = 2, and n=12n = 12: t12=1.5×(2)12−1⇒t12=1.5×(2)11⇒t12=1.5×2048⇒t12=3072t_{12} = 1.5 \times (2)^{12-1} \Rightarrow t_{12} = 1.5 \times (2)^{11} \Rightarrow t_{12} = 1.5 \times 2048 \Rightarrow t_{12} = 3072.

Therefore, the 12th term of the GP is 3072.

Exercise: Exercise Set 8.3 | Q: 1 | (Page 20)
For More Understanding

Explanation

The textbook context provides the general formula for the nth term of a Geometric Progression as t_n = arn−1ar^{n-1}. To find the 12th term, we first need to determine the first term 'a'. By substituting the given 8th term (192) and common ratio (2) into the formula, we solve for 'a'. Once 'a' is found, we substitute it back into the formula along with n=12 and r=2 to calculate the 12th term.

Solution Steps

  1. Step 1: Write the formula for the nth term of a GP: t_n = arn−1ar^{n-1}.

  2. Step 2: Find the first term 'a' using the 8th term: 192 = a(2)^{8-1} => 192 = 128a => a = 1.5.

  3. Step 3: Calculate the 12th term: t_{12} = 1.5 * (2)^{12-1} = 1.5 * 2048 = 3072.

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