Question

How many terms of the series 1 + 4 + 16 + ... make the sum 1365?​

Answer 1

Answer:

6 terms

Step-by-step explanation:

Given:

1+4+16+...  & S=1365

We have here GP with 1st term of 1 and common factor 4 and sum of n terms equal to 1365.

Sum of n terms:

Sₙ=a(rⁿ-1)/(r-1), where a=1, r=4

• a(rⁿ-1)/(r-1)=1365
• (4ⁿ-1)/(4-1)=1365
• 4ⁿ-1=1365*3
• 4ⁿ=4095+1
• 4ⁿ=4096
• n=6

Number of terms = 6

Answer 2

6 terms make the sum 1365 of series 1+4+16+...

Given:

• A series
• 1+4+16+...

To find:

• How many terms of the series 1 + 4 + 16 + ... make the sum 1365?

Solution:

Formula to be used:

Sum of first n terms of GP: $\bf S_n = \frac{a({r}^{n} - 1) }{r - 1}$

here,

r is common ratio of G.P.

Step 1:

It has been clear that given series forms a G.P.

here,

1 st term (a)= 1

Common ratio (r)=4

Sum of n terms =1365

Step 2:

Find the value of n.

Put all the values in the formula of sum .

$1365 = \frac{1({4}^{n} - 1) }{4 - 1}$

or

$1365 = \frac{({4}^{n} - 1) }{3}$

or

${4}^{n} - 1 = 1365 \times 3$

or

${4}^{n} - 1 = 4095$

or

${4}^{n} = 4096$

Step 3:

Write 4096 in powers of 4 and equate powers.

${4}^{n} = {4}^{6}$

when base are same we can compare powers.

So,

$\bf n = 6$

Thus,

6 terms make the sum 1365 of series 1+4+16+...

Learn more:

1) find five numbers in GP such that their sum is 31/4 and product is 1

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2) If the first term of a G.P is 16 and its sum to infinity is 96/17 then find the common ratio

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