**Find the Terms From Sum and product of 3 Consecutive Terms of GP :**

In this section, we will learn how to find the terms from sum and product of 3 consecutive terms of a geometric progression.

We can consider the first three terms of a geometric progression as

a/r, a, ar

Let us see some examples based on the concept.

**Example 1 :**

If the product of three consecutive terms in G.P is 216 the sum of their product in pairs is 156, find them.

**Solution :**

Let the first three terms are a/r, a, ar

Product of three terms = 216

(a/r) ⋅ a ⋅ a r = 216

a^{3} = 6^{3}

a = 6

Sum of their product in pairs = 156

(a/r) ⋅ a + a ⋅ ar + ar ⋅ (a/r) = 156

a^{2} / r + a^{2} r + a^{2} = 156

a^{2} [ (1/r) + r + 1 ] = 156

a² [ (1+r²+r)/r] = 156

a² (r²+r+1)/r = 156

(6²/r)(r²+r+1) = 156

(r²+r+1)/r = 156/36

(r²+r+1)/r = 13/3

3(r²+r+1) = 13 r

3r² + 3r + 3 - 13r = 0

3r² - 10r + 3 = 0

(3r - 1)(r - 3) = 0

3r - 1 = 0 3r = 1 r = 1/3 |
r - 3 = 0 r = 3 |

If a = 6, then r = 1/3

a/r = 6/(1/3) ==> 18

a = 6

ar = 6(1/3) ==> 2

If a = 6, then r = 1/3

a/r = 6/3 ==> 2

a = 6

ar = 6(3) ==> 18

Hence the required three terms are 18, 6 and 2 or 2, 6, 18.

**Example 2 :**

Find the first three consecutive terms in G.P whose sum is 7 and the sum of their reciprocals is 7/4.

**Solution :**

Let the first three terms are a/r, a, ar

Sum of three terms = 7

a/r + a + a r = 7

a [(1/r) + 1 + r] = 7

[(1/r) + 1 + r] = 7/a -----------(1)

Sum of their reciprocals = 7/4

(r/a) + (1/a) + (1/ar) = 7/4

(1/a)[r + 1 + (1/r)] = 7/4

(1/a)(7/a) = 7/4

7/a² = 7/4

a^{2} = 4

a = 2

By applying a = 2 in the in (1), we get

[(1/r) + 1 + r] = 7/2

(1 + r + r^{2})/r = 7/2

2r^{2} + 2r + 2 = 7r

2r² + 2r - 7r + 2 = 0

2r² - 5r + 2 = 0

(2r - 1) (r - 2) = 0

2r - 1 = 0 r = 1/2 |
r - 2 = 0 r = 2 |

If a = 2 and r = 1/2

a/r = 2/(1/2) ==> 4

a = 2

ar = 2(1/2) ==> 1

Therefore the three terms are 4, 2 , 1 or 1 , 2 , 4

After having gone through the stuff given above, we hope that the students would have understood how to find the terms from the sum and product of 3 terms of geometric progression.

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